diff --git a/include/pybind11/cast.h b/include/pybind11/cast.h
index 4708101d80..556bdb7e34 100644
--- a/include/pybind11/cast.h
+++ b/include/pybind11/cast.h
@@ -347,9 +347,12 @@ struct type_caster<T, enable_if_t<std::is_arithmetic<T>::value && !is_std_char_t
         return PyLong_FromUnsignedLongLong((unsigned long long) src);
     }
 
-    PYBIND11_TYPE_CASTER(T,
-                         io_name<std::is_integral<T>::value>(
-                             "typing.SupportsInt", "int", "typing.SupportsFloat", "float"));
+    PYBIND11_TYPE_CASTER(
+        T,
+        io_name<std::is_integral<T>::value>("typing.SupportsInt | typing.SupportsIndex",
+                                            "int",
+                                            "typing.SupportsFloat | typing.SupportsIndex",
+                                            "float"));
 };
 
 template <typename T>
diff --git a/include/pybind11/complex.h b/include/pybind11/complex.h
index 8a831c12ce..0b6f49365d 100644
--- a/include/pybind11/complex.h
+++ b/include/pybind11/complex.h
@@ -54,7 +54,23 @@ class type_caster<std::complex<T>> {
         if (!convert && !PyComplex_Check(src.ptr())) {
             return false;
         }
-        Py_complex result = PyComplex_AsCComplex(src.ptr());
+        handle src_or_index = src;
+        // PyPy: 7.3.7's 3.8 does not implement PyLong_*'s __index__ calls.
+        // The same logic is used in numeric_caster for ints and floats
+#if defined(PYPY_VERSION)
+        object index;
+        if (PYBIND11_INDEX_CHECK(src.ptr())) {
+            index = reinterpret_steal<object>(PyNumber_Index(src.ptr()));
+            if (!index) {
+                PyErr_Clear();
+                if (!convert)
+                    return false;
+            } else {
+                src_or_index = index;
+            }
+        }
+#endif
+        Py_complex result = PyComplex_AsCComplex(src_or_index.ptr());
         if (result.real == -1.0 && PyErr_Occurred()) {
             PyErr_Clear();
             return false;
@@ -68,7 +84,10 @@ class type_caster<std::complex<T>> {
         return PyComplex_FromDoubles((double) src.real(), (double) src.imag());
     }
 
-    PYBIND11_TYPE_CASTER(std::complex<T>, const_name("complex"));
+    PYBIND11_TYPE_CASTER(
+        std::complex<T>,
+        io_name("typing.SupportsComplex | typing.SupportsFloat | typing.SupportsIndex",
+                "complex"));
 };
 PYBIND11_NAMESPACE_END(detail)
 PYBIND11_NAMESPACE_END(PYBIND11_NAMESPACE)
diff --git a/include/pybind11/detail/common.h b/include/pybind11/detail/common.h
index 16952c5829..07c0943006 100644
--- a/include/pybind11/detail/common.h
+++ b/include/pybind11/detail/common.h
@@ -322,6 +322,13 @@
 #define PYBIND11_BYTES_AS_STRING PyBytes_AsString
 #define PYBIND11_BYTES_SIZE PyBytes_Size
 #define PYBIND11_LONG_CHECK(o) PyLong_Check(o)
+// In PyPy 7.3.3, `PyIndex_Check` is implemented by calling `__index__`,
+// while CPython only considers the existence of `nb_index`/`__index__`.
+#if !defined(PYPY_VERSION)
+#    define PYBIND11_INDEX_CHECK(o) PyIndex_Check(o)
+#else
+#    define PYBIND11_INDEX_CHECK(o) hasattr(o, "__index__")
+#endif
 #define PYBIND11_LONG_AS_LONGLONG(o) PyLong_AsLongLong(o)
 #define PYBIND11_LONG_FROM_SIGNED(o) PyLong_FromSsize_t((ssize_t) (o))
 #define PYBIND11_LONG_FROM_UNSIGNED(o) PyLong_FromSize_t((size_t) (o))
diff --git a/tests/test_builtin_casters.cpp b/tests/test_builtin_casters.cpp
index c516f8de7e..1aa9f89b42 100644
--- a/tests/test_builtin_casters.cpp
+++ b/tests/test_builtin_casters.cpp
@@ -363,6 +363,8 @@ TEST_SUBMODULE(builtin_casters, m) {
     m.def("complex_cast", [](float x) { return "{}"_s.format(x); });
     m.def("complex_cast",
           [](std::complex<float> x) { return "({}, {})"_s.format(x.real(), x.imag()); });
+    m.def("complex_convert", [](std::complex<float> x) { return x; });
+    m.def("complex_noconvert", [](std::complex<float> x) { return x; }, py::arg{}.noconvert());
 
     // test int vs. long (Python 2)
     m.def("int_cast", []() { return (int) 42; });
diff --git a/tests/test_builtin_casters.py b/tests/test_builtin_casters.py
index 7a2c6a4d8f..ae1b1bd179 100644
--- a/tests/test_builtin_casters.py
+++ b/tests/test_builtin_casters.py
@@ -286,7 +286,10 @@ def __int__(self):
 
     convert, noconvert = m.int_passthrough, m.int_passthrough_noconvert
 
-    assert doc(convert) == "int_passthrough(arg0: typing.SupportsInt) -> int"
+    assert (
+        doc(convert)
+        == "int_passthrough(arg0: typing.SupportsInt | typing.SupportsIndex) -> int"
+    )
     assert doc(noconvert) == "int_passthrough_noconvert(arg0: int) -> int"
 
     def requires_conversion(v):
@@ -322,19 +325,39 @@ def cant_convert(v):
 
 
 def test_float_convert(doc):
+    class Int:
+        def __int__(self):
+            return -5
+
+    class Index:
+        def __index__(self) -> int:
+            return -7
+
     class Float:
         def __float__(self):
             return 41.45
 
     convert, noconvert = m.float_passthrough, m.float_passthrough_noconvert
-    assert doc(convert) == "float_passthrough(arg0: typing.SupportsFloat) -> float"
+    assert (
+        doc(convert)
+        == "float_passthrough(arg0: typing.SupportsFloat | typing.SupportsIndex) -> float"
+    )
     assert doc(noconvert) == "float_passthrough_noconvert(arg0: float) -> float"
 
     def requires_conversion(v):
         pytest.raises(TypeError, noconvert, v)
 
+    def cant_convert(v):
+        pytest.raises(TypeError, convert, v)
+
     requires_conversion(Float())
+    requires_conversion(Index())
     assert pytest.approx(convert(Float())) == 41.45
+    assert pytest.approx(convert(Index())) == -7.0
+    assert isinstance(convert(Float()), float)
+    assert pytest.approx(convert(3)) == 3.0
+    requires_conversion(3)
+    cant_convert(Int())
 
 
 def test_numpy_int_convert():
@@ -381,7 +404,7 @@ def test_tuple(doc):
     assert (
         doc(m.tuple_passthrough)
         == """
-        tuple_passthrough(arg0: tuple[bool, str, typing.SupportsInt]) -> tuple[int, str, bool]
+        tuple_passthrough(arg0: tuple[bool, str, typing.SupportsInt | typing.SupportsIndex]) -> tuple[int, str, bool]
 
         Return a triple in reversed order
     """
@@ -458,11 +481,61 @@ def test_reference_wrapper():
     assert m.refwrap_call_iiw(IncType(10), m.refwrap_iiw) == [10, 10, 10, 10]
 
 
-def test_complex_cast():
+def test_complex_cast(doc):
     """std::complex casts"""
+
+    class Complex:
+        def __complex__(self) -> complex:
+            return complex(5, 4)
+
+    class Float:
+        def __float__(self) -> float:
+            return 5.0
+
+    class Int:
+        def __int__(self) -> int:
+            return 3
+
+    class Index:
+        def __index__(self) -> int:
+            return 1
+
     assert m.complex_cast(1) == "1.0"
+    assert m.complex_cast(1.0) == "1.0"
+    assert m.complex_cast(Complex()) == "(5.0, 4.0)"
     assert m.complex_cast(2j) == "(0.0, 2.0)"
 
+    convert, noconvert = m.complex_convert, m.complex_noconvert
+
+    def requires_conversion(v):
+        pytest.raises(TypeError, noconvert, v)
+
+    def cant_convert(v):
+        pytest.raises(TypeError, convert, v)
+
+    assert (
+        doc(convert)
+        == "complex_convert(arg0: typing.SupportsComplex | typing.SupportsFloat | typing.SupportsIndex) -> complex"
+    )
+    assert doc(noconvert) == "complex_noconvert(arg0: complex) -> complex"
+
+    assert convert(1) == 1.0
+    assert convert(2.0) == 2.0
+    assert convert(1 + 5j) == 1.0 + 5.0j
+    assert convert(Complex()) == 5.0 + 4j
+    assert convert(Float()) == 5.0
+    assert isinstance(convert(Float()), complex)
+    cant_convert(Int())
+    assert convert(Index()) == 1
+    assert isinstance(convert(Index()), complex)
+
+    requires_conversion(1)
+    requires_conversion(2.0)
+    assert noconvert(1 + 5j) == 1.0 + 5.0j
+    requires_conversion(Complex())
+    requires_conversion(Float())
+    requires_conversion(Index())
+
 
 def test_bool_caster():
     """Test bool caster implicit conversions."""
diff --git a/tests/test_callbacks.py b/tests/test_callbacks.py
index 09dadd94c3..c0a57a7b86 100644
--- a/tests/test_callbacks.py
+++ b/tests/test_callbacks.py
@@ -140,11 +140,11 @@ def test_cpp_function_roundtrip():
 def test_function_signatures(doc):
     assert (
         doc(m.test_callback3)
-        == "test_callback3(arg0: collections.abc.Callable[[typing.SupportsInt], int]) -> str"
+        == "test_callback3(arg0: collections.abc.Callable[[typing.SupportsInt | typing.SupportsIndex], int]) -> str"
     )
     assert (
         doc(m.test_callback4)
-        == "test_callback4() -> collections.abc.Callable[[typing.SupportsInt], int]"
+        == "test_callback4() -> collections.abc.Callable[[typing.SupportsInt | typing.SupportsIndex], int]"
     )
 
 
diff --git a/tests/test_class.py b/tests/test_class.py
index 1e82930361..fae6a31899 100644
--- a/tests/test_class.py
+++ b/tests/test_class.py
@@ -163,13 +163,13 @@ def test_qualname(doc):
     assert (
         doc(m.NestBase.Nested.fn)
         == """
-        fn(self: m.class_.NestBase.Nested, arg0: typing.SupportsInt, arg1: m.class_.NestBase, arg2: m.class_.NestBase.Nested) -> None
+        fn(self: m.class_.NestBase.Nested, arg0: typing.SupportsInt | typing.SupportsIndex, arg1: m.class_.NestBase, arg2: m.class_.NestBase.Nested) -> None
     """
     )
     assert (
         doc(m.NestBase.Nested.fa)
         == """
-        fa(self: m.class_.NestBase.Nested, a: typing.SupportsInt, b: m.class_.NestBase, c: m.class_.NestBase.Nested) -> None
+        fa(self: m.class_.NestBase.Nested, a: typing.SupportsInt | typing.SupportsIndex, b: m.class_.NestBase, c: m.class_.NestBase.Nested) -> None
     """
     )
     assert m.NestBase.__module__ == "pybind11_tests.class_"
diff --git a/tests/test_custom_type_casters.py b/tests/test_custom_type_casters.py
index bf31d3f374..6ed1c564f0 100644
--- a/tests/test_custom_type_casters.py
+++ b/tests/test_custom_type_casters.py
@@ -75,7 +75,7 @@ def test_noconvert_args(msg):
         msg(excinfo.value)
         == """
         ints_preferred(): incompatible function arguments. The following argument types are supported:
-            1. (i: typing.SupportsInt) -> int
+            1. (i: typing.SupportsInt | typing.SupportsIndex) -> int
 
         Invoked with: 4.0
     """
diff --git a/tests/test_docstring_options.py b/tests/test_docstring_options.py
index f2a10480ca..802a1ec9e5 100644
--- a/tests/test_docstring_options.py
+++ b/tests/test_docstring_options.py
@@ -20,11 +20,11 @@ def test_docstring_options():
 
     # options.enable_function_signatures()
     assert m.test_function3.__doc__.startswith(
-        "test_function3(a: typing.SupportsInt, b: typing.SupportsInt) -> None"
+        "test_function3(a: typing.SupportsInt | typing.SupportsIndex, b: typing.SupportsInt | typing.SupportsIndex) -> None"
     )
 
     assert m.test_function4.__doc__.startswith(
-        "test_function4(a: typing.SupportsInt, b: typing.SupportsInt) -> None"
+        "test_function4(a: typing.SupportsInt | typing.SupportsIndex, b: typing.SupportsInt | typing.SupportsIndex) -> None"
     )
     assert m.test_function4.__doc__.endswith("A custom docstring\n")
 
@@ -37,7 +37,7 @@ def test_docstring_options():
 
     # RAII destructor
     assert m.test_function7.__doc__.startswith(
-        "test_function7(a: typing.SupportsInt, b: typing.SupportsInt) -> None"
+        "test_function7(a: typing.SupportsInt | typing.SupportsIndex, b: typing.SupportsInt | typing.SupportsIndex) -> None"
     )
     assert m.test_function7.__doc__.endswith("A custom docstring\n")
 
diff --git a/tests/test_enum.py b/tests/test_enum.py
index 99d4a88c8a..f295b01457 100644
--- a/tests/test_enum.py
+++ b/tests/test_enum.py
@@ -328,7 +328,7 @@ def test_generated_dunder_methods_pos_only():
         )
         assert (
             re.match(
-                r"^__setstate__\(self: [\w\.]+, state: [\w\.]+, /\)",
+                r"^__setstate__\(self: [\w\.]+, state: [\w\. \|]+, /\)",
                 enum_type.__setstate__.__doc__,
             )
             is not None
diff --git a/tests/test_factory_constructors.py b/tests/test_factory_constructors.py
index 67f859b9ab..c6ae98c7fb 100644
--- a/tests/test_factory_constructors.py
+++ b/tests/test_factory_constructors.py
@@ -78,10 +78,10 @@ def test_init_factory_signature(msg):
         msg(excinfo.value)
         == """
         __init__(): incompatible constructor arguments. The following argument types are supported:
-            1. m.factory_constructors.TestFactory1(arg0: m.factory_constructors.tag.unique_ptr_tag, arg1: typing.SupportsInt)
+            1. m.factory_constructors.TestFactory1(arg0: m.factory_constructors.tag.unique_ptr_tag, arg1: typing.SupportsInt | typing.SupportsIndex)
             2. m.factory_constructors.TestFactory1(arg0: str)
             3. m.factory_constructors.TestFactory1(arg0: m.factory_constructors.tag.pointer_tag)
-            4. m.factory_constructors.TestFactory1(arg0: object, arg1: typing.SupportsInt, arg2: object)
+            4. m.factory_constructors.TestFactory1(arg0: object, arg1: typing.SupportsInt | typing.SupportsIndex, arg2: object)
 
         Invoked with: 'invalid', 'constructor', 'arguments'
     """
@@ -93,13 +93,13 @@ def test_init_factory_signature(msg):
         __init__(*args, **kwargs)
         Overloaded function.
 
-        1. __init__(self: m.factory_constructors.TestFactory1, arg0: m.factory_constructors.tag.unique_ptr_tag, arg1: typing.SupportsInt) -> None
+        1. __init__(self: m.factory_constructors.TestFactory1, arg0: m.factory_constructors.tag.unique_ptr_tag, arg1: typing.SupportsInt | typing.SupportsIndex) -> None
 
         2. __init__(self: m.factory_constructors.TestFactory1, arg0: str) -> None
 
         3. __init__(self: m.factory_constructors.TestFactory1, arg0: m.factory_constructors.tag.pointer_tag) -> None
 
-        4. __init__(self: m.factory_constructors.TestFactory1, arg0: object, arg1: typing.SupportsInt, arg2: object) -> None
+        4. __init__(self: m.factory_constructors.TestFactory1, arg0: object, arg1: typing.SupportsInt | typing.SupportsIndex, arg2: object) -> None
     """
     )
 
diff --git a/tests/test_kwargs_and_defaults.py b/tests/test_kwargs_and_defaults.py
index b62e4b7412..57345f128f 100644
--- a/tests/test_kwargs_and_defaults.py
+++ b/tests/test_kwargs_and_defaults.py
@@ -9,28 +9,28 @@
 def test_function_signatures(doc):
     assert (
         doc(m.kw_func0)
-        == "kw_func0(arg0: typing.SupportsInt, arg1: typing.SupportsInt) -> str"
+        == "kw_func0(arg0: typing.SupportsInt | typing.SupportsIndex, arg1: typing.SupportsInt | typing.SupportsIndex) -> str"
     )
     assert (
         doc(m.kw_func1)
-        == "kw_func1(x: typing.SupportsInt, y: typing.SupportsInt) -> str"
+        == "kw_func1(x: typing.SupportsInt | typing.SupportsIndex, y: typing.SupportsInt | typing.SupportsIndex) -> str"
     )
     assert (
         doc(m.kw_func2)
-        == "kw_func2(x: typing.SupportsInt = 100, y: typing.SupportsInt = 200) -> str"
+        == "kw_func2(x: typing.SupportsInt | typing.SupportsIndex = 100, y: typing.SupportsInt | typing.SupportsIndex = 200) -> str"
     )
     assert doc(m.kw_func3) == "kw_func3(data: str = 'Hello world!') -> None"
     assert (
         doc(m.kw_func4)
-        == "kw_func4(myList: collections.abc.Sequence[typing.SupportsInt] = [13, 17]) -> str"
+        == "kw_func4(myList: collections.abc.Sequence[typing.SupportsInt | typing.SupportsIndex] = [13, 17]) -> str"
     )
     assert (
         doc(m.kw_func_udl)
-        == "kw_func_udl(x: typing.SupportsInt, y: typing.SupportsInt = 300) -> str"
+        == "kw_func_udl(x: typing.SupportsInt | typing.SupportsIndex, y: typing.SupportsInt | typing.SupportsIndex = 300) -> str"
     )
     assert (
         doc(m.kw_func_udl_z)
-        == "kw_func_udl_z(x: typing.SupportsInt, y: typing.SupportsInt = 0) -> str"
+        == "kw_func_udl_z(x: typing.SupportsInt | typing.SupportsIndex, y: typing.SupportsInt | typing.SupportsIndex = 0) -> str"
     )
     assert doc(m.args_function) == "args_function(*args) -> tuple"
     assert (
@@ -42,11 +42,11 @@ def test_function_signatures(doc):
     )
     assert (
         doc(m.KWClass.foo0)
-        == "foo0(self: m.kwargs_and_defaults.KWClass, arg0: typing.SupportsInt, arg1: typing.SupportsFloat) -> None"
+        == "foo0(self: m.kwargs_and_defaults.KWClass, arg0: typing.SupportsInt | typing.SupportsIndex, arg1: typing.SupportsFloat | typing.SupportsIndex) -> None"
     )
     assert (
         doc(m.KWClass.foo1)
-        == "foo1(self: m.kwargs_and_defaults.KWClass, x: typing.SupportsInt, y: typing.SupportsFloat) -> None"
+        == "foo1(self: m.kwargs_and_defaults.KWClass, x: typing.SupportsInt | typing.SupportsIndex, y: typing.SupportsFloat | typing.SupportsIndex) -> None"
     )
     assert (
         doc(m.kw_lb_func0)
@@ -138,7 +138,7 @@ def test_mixed_args_and_kwargs(msg):
         msg(excinfo.value)
         == """
         mixed_plus_args(): incompatible function arguments. The following argument types are supported:
-            1. (arg0: typing.SupportsInt, arg1: typing.SupportsFloat, *args) -> tuple
+            1. (arg0: typing.SupportsInt | typing.SupportsIndex, arg1: typing.SupportsFloat | typing.SupportsIndex, *args) -> tuple
 
         Invoked with: 1
     """
@@ -149,7 +149,7 @@ def test_mixed_args_and_kwargs(msg):
         msg(excinfo.value)
         == """
         mixed_plus_args(): incompatible function arguments. The following argument types are supported:
-            1. (arg0: typing.SupportsInt, arg1: typing.SupportsFloat, *args) -> tuple
+            1. (arg0: typing.SupportsInt | typing.SupportsIndex, arg1: typing.SupportsFloat | typing.SupportsIndex, *args) -> tuple
 
         Invoked with:
     """
@@ -183,7 +183,7 @@ def test_mixed_args_and_kwargs(msg):
         msg(excinfo.value)
         == """
         mixed_plus_args_kwargs_defaults(): incompatible function arguments. The following argument types are supported:
-            1. (i: typing.SupportsInt = 1, j: typing.SupportsFloat = 3.14159, *args, **kwargs) -> tuple
+            1. (i: typing.SupportsInt | typing.SupportsIndex = 1, j: typing.SupportsFloat | typing.SupportsIndex = 3.14159, *args, **kwargs) -> tuple
 
         Invoked with: 1; kwargs: i=1
     """
@@ -194,7 +194,7 @@ def test_mixed_args_and_kwargs(msg):
         msg(excinfo.value)
         == """
         mixed_plus_args_kwargs_defaults(): incompatible function arguments. The following argument types are supported:
-            1. (i: typing.SupportsInt = 1, j: typing.SupportsFloat = 3.14159, *args, **kwargs) -> tuple
+            1. (i: typing.SupportsInt | typing.SupportsIndex = 1, j: typing.SupportsFloat | typing.SupportsIndex = 3.14159, *args, **kwargs) -> tuple
 
         Invoked with: 1, 2; kwargs: j=1
     """
@@ -211,7 +211,7 @@ def test_mixed_args_and_kwargs(msg):
         msg(excinfo.value)
         == """
         args_kwonly(): incompatible function arguments. The following argument types are supported:
-            1. (i: typing.SupportsInt, j: typing.SupportsFloat, *args, z: typing.SupportsInt) -> tuple
+            1. (i: typing.SupportsInt | typing.SupportsIndex, j: typing.SupportsFloat | typing.SupportsIndex, *args, z: typing.SupportsInt | typing.SupportsIndex) -> tuple
 
         Invoked with: 2, 2.5, 22
     """
@@ -233,12 +233,12 @@ def test_mixed_args_and_kwargs(msg):
     )
     assert (
         m.args_kwonly_kwargs.__doc__
-        == "args_kwonly_kwargs(i: typing.SupportsInt, j: typing.SupportsFloat, *args, z: typing.SupportsInt, **kwargs) -> tuple\n"
+        == "args_kwonly_kwargs(i: typing.SupportsInt | typing.SupportsIndex, j: typing.SupportsFloat | typing.SupportsIndex, *args, z: typing.SupportsInt | typing.SupportsIndex, **kwargs) -> tuple\n"
     )
 
     assert (
         m.args_kwonly_kwargs_defaults.__doc__
-        == "args_kwonly_kwargs_defaults(i: typing.SupportsInt = 1, j: typing.SupportsFloat = 3.14159, *args, z: typing.SupportsInt = 42, **kwargs) -> tuple\n"
+        == "args_kwonly_kwargs_defaults(i: typing.SupportsInt | typing.SupportsIndex = 1, j: typing.SupportsFloat | typing.SupportsIndex = 3.14159, *args, z: typing.SupportsInt | typing.SupportsIndex = 42, **kwargs) -> tuple\n"
     )
     assert m.args_kwonly_kwargs_defaults() == (1, 3.14159, (), 42, {})
     assert m.args_kwonly_kwargs_defaults(2) == (2, 3.14159, (), 42, {})
@@ -294,11 +294,11 @@ def test_keyword_only_args(msg):
     x.method(i=1, j=2)
     assert (
         m.first_arg_kw_only.__init__.__doc__
-        == "__init__(self: pybind11_tests.kwargs_and_defaults.first_arg_kw_only, *, i: typing.SupportsInt = 0) -> None\n"
+        == "__init__(self: pybind11_tests.kwargs_and_defaults.first_arg_kw_only, *, i: typing.SupportsInt | typing.SupportsIndex = 0) -> None\n"
     )
     assert (
         m.first_arg_kw_only.method.__doc__
-        == "method(self: pybind11_tests.kwargs_and_defaults.first_arg_kw_only, *, i: typing.SupportsInt = 1, j: typing.SupportsInt = 2) -> None\n"
+        == "method(self: pybind11_tests.kwargs_and_defaults.first_arg_kw_only, *, i: typing.SupportsInt | typing.SupportsIndex = 1, j: typing.SupportsInt | typing.SupportsIndex = 2) -> None\n"
     )
 
 
@@ -344,7 +344,7 @@ def test_positional_only_args():
     # Mix it with args and kwargs:
     assert (
         m.args_kwonly_full_monty.__doc__
-        == "args_kwonly_full_monty(arg0: typing.SupportsInt = 1, arg1: typing.SupportsInt = 2, /, j: typing.SupportsFloat = 3.14159, *args, z: typing.SupportsInt = 42, **kwargs) -> tuple\n"
+        == "args_kwonly_full_monty(arg0: typing.SupportsInt | typing.SupportsIndex = 1, arg1: typing.SupportsInt | typing.SupportsIndex = 2, /, j: typing.SupportsFloat | typing.SupportsIndex = 3.14159, *args, z: typing.SupportsInt | typing.SupportsIndex = 42, **kwargs) -> tuple\n"
     )
     assert m.args_kwonly_full_monty() == (1, 2, 3.14159, (), 42, {})
     assert m.args_kwonly_full_monty(8) == (8, 2, 3.14159, (), 42, {})
@@ -387,30 +387,30 @@ def test_positional_only_args():
     # https://github.com/pybind/pybind11/pull/3402#issuecomment-963341987
     assert (
         m.first_arg_kw_only.pos_only.__doc__
-        == "pos_only(self: pybind11_tests.kwargs_and_defaults.first_arg_kw_only, /, i: typing.SupportsInt, j: typing.SupportsInt) -> None\n"
+        == "pos_only(self: pybind11_tests.kwargs_and_defaults.first_arg_kw_only, /, i: typing.SupportsInt | typing.SupportsIndex, j: typing.SupportsInt | typing.SupportsIndex) -> None\n"
     )
 
 
 def test_signatures():
     assert (
         m.kw_only_all.__doc__
-        == "kw_only_all(*, i: typing.SupportsInt, j: typing.SupportsInt) -> tuple\n"
+        == "kw_only_all(*, i: typing.SupportsInt | typing.SupportsIndex, j: typing.SupportsInt | typing.SupportsIndex) -> tuple\n"
     )
     assert (
         m.kw_only_mixed.__doc__
-        == "kw_only_mixed(i: typing.SupportsInt, *, j: typing.SupportsInt) -> tuple\n"
+        == "kw_only_mixed(i: typing.SupportsInt | typing.SupportsIndex, *, j: typing.SupportsInt | typing.SupportsIndex) -> tuple\n"
     )
     assert (
         m.pos_only_all.__doc__
-        == "pos_only_all(i: typing.SupportsInt, j: typing.SupportsInt, /) -> tuple\n"
+        == "pos_only_all(i: typing.SupportsInt | typing.SupportsIndex, j: typing.SupportsInt | typing.SupportsIndex, /) -> tuple\n"
     )
     assert (
         m.pos_only_mix.__doc__
-        == "pos_only_mix(i: typing.SupportsInt, /, j: typing.SupportsInt) -> tuple\n"
+        == "pos_only_mix(i: typing.SupportsInt | typing.SupportsIndex, /, j: typing.SupportsInt | typing.SupportsIndex) -> tuple\n"
     )
     assert (
         m.pos_kw_only_mix.__doc__
-        == "pos_kw_only_mix(i: typing.SupportsInt, /, j: typing.SupportsInt, *, k: typing.SupportsInt) -> tuple\n"
+        == "pos_kw_only_mix(i: typing.SupportsInt | typing.SupportsIndex, /, j: typing.SupportsInt | typing.SupportsIndex, *, k: typing.SupportsInt | typing.SupportsIndex) -> tuple\n"
     )
 
 
diff --git a/tests/test_methods_and_attributes.py b/tests/test_methods_and_attributes.py
index 6a8d993cb6..553d5bfc1b 100644
--- a/tests/test_methods_and_attributes.py
+++ b/tests/test_methods_and_attributes.py
@@ -251,7 +251,7 @@ def test_no_mixed_overloads():
             "#define PYBIND11_DETAILED_ERROR_MESSAGES or compile in debug mode for more details"
             if not detailed_error_messages_enabled
             else "error while attempting to bind static method ExampleMandA.overload_mixed1"
-            "(arg0: typing.SupportsFloat) -> str"
+            "(arg0: typing.SupportsFloat | typing.SupportsIndex) -> str"
         )
     )
 
@@ -264,7 +264,7 @@ def test_no_mixed_overloads():
             "#define PYBIND11_DETAILED_ERROR_MESSAGES or compile in debug mode for more details"
             if not detailed_error_messages_enabled
             else "error while attempting to bind instance method ExampleMandA.overload_mixed2"
-            "(self: pybind11_tests.methods_and_attributes.ExampleMandA, arg0: typing.SupportsInt, arg1: typing.SupportsInt)"
+            "(self: pybind11_tests.methods_and_attributes.ExampleMandA, arg0: typing.SupportsInt | typing.SupportsIndex, arg1: typing.SupportsInt | typing.SupportsIndex)"
             " -> str"
         )
     )
@@ -491,7 +491,7 @@ def test_str_issue(msg):
         msg(excinfo.value)
         == """
         __init__(): incompatible constructor arguments. The following argument types are supported:
-            1. m.methods_and_attributes.StrIssue(arg0: typing.SupportsInt)
+            1. m.methods_and_attributes.StrIssue(arg0: typing.SupportsInt | typing.SupportsIndex)
             2. m.methods_and_attributes.StrIssue()
 
         Invoked with: 'no', 'such', 'constructor'
@@ -534,21 +534,27 @@ def test_overload_ordering():
     assert m.overload_order(0) == 4
 
     assert (
-        "1. overload_order(arg0: typing.SupportsInt) -> int" in m.overload_order.__doc__
+        "1. overload_order(arg0: typing.SupportsInt | typing.SupportsIndex) -> int"
+        in m.overload_order.__doc__
     )
     assert "2. overload_order(arg0: str) -> int" in m.overload_order.__doc__
     assert "3. overload_order(arg0: str) -> int" in m.overload_order.__doc__
     assert (
-        "4. overload_order(arg0: typing.SupportsInt) -> int" in m.overload_order.__doc__
+        "4. overload_order(arg0: typing.SupportsInt | typing.SupportsIndex) -> int"
+        in m.overload_order.__doc__
     )
 
     with pytest.raises(TypeError) as err:
         m.overload_order(1.1)
 
-    assert "1. (arg0: typing.SupportsInt) -> int" in str(err.value)
+    assert "1. (arg0: typing.SupportsInt | typing.SupportsIndex) -> int" in str(
+        err.value
+    )
     assert "2. (arg0: str) -> int" in str(err.value)
     assert "3. (arg0: str) -> int" in str(err.value)
-    assert "4. (arg0: typing.SupportsInt) -> int" in str(err.value)
+    assert "4. (arg0: typing.SupportsInt | typing.SupportsIndex) -> int" in str(
+        err.value
+    )
 
 
 def test_rvalue_ref_param():
diff --git a/tests/test_numpy_dtypes.py b/tests/test_numpy_dtypes.py
index 9f85742809..22814aba5a 100644
--- a/tests/test_numpy_dtypes.py
+++ b/tests/test_numpy_dtypes.py
@@ -367,7 +367,7 @@ def test_complex_array():
 def test_signature(doc):
     assert (
         doc(m.create_rec_nested)
-        == "create_rec_nested(arg0: typing.SupportsInt) -> numpy.typing.NDArray[NestedStruct]"
+        == "create_rec_nested(arg0: typing.SupportsInt | typing.SupportsIndex) -> numpy.typing.NDArray[NestedStruct]"
     )
 
 
diff --git a/tests/test_numpy_vectorize.py b/tests/test_numpy_vectorize.py
index d405e68002..05f7c704f5 100644
--- a/tests/test_numpy_vectorize.py
+++ b/tests/test_numpy_vectorize.py
@@ -211,11 +211,11 @@ def test_passthrough_arguments(doc):
         "vec_passthrough("
         + ", ".join(
             [
-                "arg0: typing.SupportsFloat",
+                "arg0: typing.SupportsFloat | typing.SupportsIndex",
                 "arg1: typing.Annotated[numpy.typing.ArrayLike, numpy.float64]",
                 "arg2: typing.Annotated[numpy.typing.ArrayLike, numpy.float64]",
                 "arg3: typing.Annotated[numpy.typing.ArrayLike, numpy.int32]",
-                "arg4: typing.SupportsInt",
+                "arg4: typing.SupportsInt | typing.SupportsIndex",
                 "arg5: m.numpy_vectorize.NonPODClass",
                 "arg6: typing.Annotated[numpy.typing.ArrayLike, numpy.float64]",
             ]
diff --git a/tests/test_pytypes.py b/tests/test_pytypes.py
index c1798f924c..09fc5f37ee 100644
--- a/tests/test_pytypes.py
+++ b/tests/test_pytypes.py
@@ -944,14 +944,14 @@ def test_tuple_variable_length_annotations(doc):
 def test_dict_annotations(doc):
     assert (
         doc(m.annotate_dict_str_int)
-        == "annotate_dict_str_int(arg0: dict[str, typing.SupportsInt]) -> None"
+        == "annotate_dict_str_int(arg0: dict[str, typing.SupportsInt | typing.SupportsIndex]) -> None"
     )
 
 
 def test_list_annotations(doc):
     assert (
         doc(m.annotate_list_int)
-        == "annotate_list_int(arg0: list[typing.SupportsInt]) -> None"
+        == "annotate_list_int(arg0: list[typing.SupportsInt | typing.SupportsIndex]) -> None"
     )
 
 
@@ -969,7 +969,7 @@ def test_iterable_annotations(doc):
 def test_iterator_annotations(doc):
     assert (
         doc(m.annotate_iterator_int)
-        == "annotate_iterator_int(arg0: collections.abc.Iterator[typing.SupportsInt]) -> None"
+        == "annotate_iterator_int(arg0: collections.abc.Iterator[typing.SupportsInt | typing.SupportsIndex]) -> None"
     )
 
 
@@ -989,7 +989,8 @@ def test_fn_return_only(doc):
 
 def test_type_annotation(doc):
     assert (
-        doc(m.annotate_type) == "annotate_type(arg0: type[typing.SupportsInt]) -> type"
+        doc(m.annotate_type)
+        == "annotate_type(arg0: type[typing.SupportsInt | typing.SupportsIndex]) -> type"
     )
 
 
@@ -1007,7 +1008,7 @@ def test_union_typing_only(doc):
 def test_union_object_annotations(doc):
     assert (
         doc(m.annotate_union_to_object)
-        == "annotate_union_to_object(arg0: typing.SupportsInt | str) -> object"
+        == "annotate_union_to_object(arg0: typing.SupportsInt | typing.SupportsIndex | str) -> object"
     )
 
 
@@ -1044,7 +1045,7 @@ def test_never_annotation(doc, backport_typehints):
 def test_optional_object_annotations(doc):
     assert (
         doc(m.annotate_optional_to_object)
-        == "annotate_optional_to_object(arg0: typing.SupportsInt | None) -> object"
+        == "annotate_optional_to_object(arg0: typing.SupportsInt | typing.SupportsIndex | None) -> object"
     )
 
 
@@ -1167,7 +1168,10 @@ def get_annotations_helper(o):
 def test_module_attribute_types() -> None:
     module_annotations = get_annotations_helper(m)
 
-    assert module_annotations["list_int"] == "list[typing.SupportsInt]"
+    assert (
+        module_annotations["list_int"]
+        == "list[typing.SupportsInt | typing.SupportsIndex]"
+    )
     assert module_annotations["set_str"] == "set[str]"
     assert module_annotations["foo"] == "pybind11_tests.pytypes.foo"
 
@@ -1190,7 +1194,10 @@ def test_get_annotations_compliance() -> None:
 
     module_annotations = get_annotations(m)
 
-    assert module_annotations["list_int"] == "list[typing.SupportsInt]"
+    assert (
+        module_annotations["list_int"]
+        == "list[typing.SupportsInt | typing.SupportsIndex]"
+    )
     assert module_annotations["set_str"] == "set[str]"
 
 
@@ -1204,10 +1211,13 @@ def test_class_attribute_types() -> None:
     instance_annotations = get_annotations_helper(m.Instance)
 
     assert empty_annotations is None
-    assert static_annotations["x"] == "typing.ClassVar[typing.SupportsFloat]"
+    assert (
+        static_annotations["x"]
+        == "typing.ClassVar[typing.SupportsFloat | typing.SupportsIndex]"
+    )
     assert (
         static_annotations["dict_str_int"]
-        == "typing.ClassVar[dict[str, typing.SupportsInt]]"
+        == "typing.ClassVar[dict[str, typing.SupportsInt | typing.SupportsIndex]]"
     )
 
     assert m.Static.x == 1.0
@@ -1219,7 +1229,7 @@ def test_class_attribute_types() -> None:
     static.dict_str_int["hi"] = 3
     assert m.Static().dict_str_int == {"hi": 3}
 
-    assert instance_annotations["y"] == "typing.SupportsFloat"
+    assert instance_annotations["y"] == "typing.SupportsFloat | typing.SupportsIndex"
     instance1 = m.Instance()
     instance1.y = 4.0
 
@@ -1236,7 +1246,10 @@ def test_class_attribute_types() -> None:
 def test_redeclaration_attr_with_type_hint() -> None:
     obj = m.Instance()
     m.attr_with_type_hint_float_x(obj)
-    assert get_annotations_helper(obj)["x"] == "typing.SupportsFloat"
+    assert (
+        get_annotations_helper(obj)["x"]
+        == "typing.SupportsFloat | typing.SupportsIndex"
+    )
     with pytest.raises(
         RuntimeError, match=r'^__annotations__\["x"\] was set already\.$'
     ):
diff --git a/tests/test_stl.py b/tests/test_stl.py
index 4a57635e27..b04f55c9f8 100644
--- a/tests/test_stl.py
+++ b/tests/test_stl.py
@@ -22,7 +22,7 @@ def test_vector(doc):
     assert doc(m.cast_vector) == "cast_vector() -> list[int]"
     assert (
         doc(m.load_vector)
-        == "load_vector(arg0: collections.abc.Sequence[typing.SupportsInt]) -> bool"
+        == "load_vector(arg0: collections.abc.Sequence[typing.SupportsInt | typing.SupportsIndex]) -> bool"
     )
 
     # Test regression caused by 936: pointers to stl containers weren't castable
@@ -51,7 +51,7 @@ def test_array(doc):
     )
     assert (
         doc(m.load_array)
-        == 'load_array(arg0: typing.Annotated[collections.abc.Sequence[typing.SupportsInt], "FixedSize(2)"]) -> bool'
+        == 'load_array(arg0: typing.Annotated[collections.abc.Sequence[typing.SupportsInt | typing.SupportsIndex], "FixedSize(2)"]) -> bool'
     )
 
 
@@ -72,7 +72,7 @@ def test_valarray(doc):
     assert doc(m.cast_valarray) == "cast_valarray() -> list[int]"
     assert (
         doc(m.load_valarray)
-        == "load_valarray(arg0: collections.abc.Sequence[typing.SupportsInt]) -> bool"
+        == "load_valarray(arg0: collections.abc.Sequence[typing.SupportsInt | typing.SupportsIndex]) -> bool"
     )
 
 
@@ -234,7 +234,7 @@ def test_reference_sensitive_optional(doc):
 
     assert (
         doc(m.double_or_zero_refsensitive)
-        == "double_or_zero_refsensitive(arg0: typing.SupportsInt | None) -> int"
+        == "double_or_zero_refsensitive(arg0: typing.SupportsInt | typing.SupportsIndex | None) -> int"
     )
 
     assert m.half_or_none_refsensitive(0) is None
@@ -352,7 +352,7 @@ def test_variant(doc):
 
     assert (
         doc(m.load_variant)
-        == "load_variant(arg0: typing.SupportsInt | str | typing.SupportsFloat | None) -> str"
+        == "load_variant(arg0: typing.SupportsInt | typing.SupportsIndex | str | typing.SupportsFloat | typing.SupportsIndex | None) -> str"
     )
 
 
@@ -368,7 +368,7 @@ def test_variant_monostate(doc):
 
     assert (
         doc(m.load_monostate_variant)
-        == "load_monostate_variant(arg0: None | typing.SupportsInt | str) -> str"
+        == "load_monostate_variant(arg0: None | typing.SupportsInt | typing.SupportsIndex | str) -> str"
     )
 
 
@@ -388,7 +388,7 @@ def test_stl_pass_by_pointer(msg):
         msg(excinfo.value)
         == """
         stl_pass_by_pointer(): incompatible function arguments. The following argument types are supported:
-            1. (v: collections.abc.Sequence[typing.SupportsInt] = None) -> list[int]
+            1. (v: collections.abc.Sequence[typing.SupportsInt | typing.SupportsIndex] = None) -> list[int]
 
         Invoked with:
     """
@@ -400,7 +400,7 @@ def test_stl_pass_by_pointer(msg):
         msg(excinfo.value)
         == """
         stl_pass_by_pointer(): incompatible function arguments. The following argument types are supported:
-            1. (v: collections.abc.Sequence[typing.SupportsInt] = None) -> list[int]
+            1. (v: collections.abc.Sequence[typing.SupportsInt | typing.SupportsIndex] = None) -> list[int]
 
         Invoked with: None
     """
@@ -615,7 +615,7 @@ class FormalSequenceLike(BareSequenceLike, Sequence):
     # convert mode
     assert (
         doc(m.roundtrip_std_vector_int)
-        == "roundtrip_std_vector_int(arg0: collections.abc.Sequence[typing.SupportsInt]) -> list[int]"
+        == "roundtrip_std_vector_int(arg0: collections.abc.Sequence[typing.SupportsInt | typing.SupportsIndex]) -> list[int]"
     )
     assert m.roundtrip_std_vector_int([1, 2, 3]) == [1, 2, 3]
     assert m.roundtrip_std_vector_int((1, 2, 3)) == [1, 2, 3]
@@ -668,7 +668,7 @@ class FormalMappingLike(BareMappingLike, Mapping):
     # convert mode
     assert (
         doc(m.roundtrip_std_map_str_int)
-        == "roundtrip_std_map_str_int(arg0: collections.abc.Mapping[str, typing.SupportsInt]) -> dict[str, int]"
+        == "roundtrip_std_map_str_int(arg0: collections.abc.Mapping[str, typing.SupportsInt | typing.SupportsIndex]) -> dict[str, int]"
     )
     assert m.roundtrip_std_map_str_int(a1b2c3) == a1b2c3
     assert m.roundtrip_std_map_str_int(FormalMappingLike(**a1b2c3)) == a1b2c3
@@ -714,7 +714,7 @@ class FormalSetLike(BareSetLike, Set):
     # convert mode
     assert (
         doc(m.roundtrip_std_set_int)
-        == "roundtrip_std_set_int(arg0: collections.abc.Set[typing.SupportsInt]) -> set[int]"
+        == "roundtrip_std_set_int(arg0: collections.abc.Set[typing.SupportsInt | typing.SupportsIndex]) -> set[int]"
     )
     assert m.roundtrip_std_set_int({1, 2, 3}) == {1, 2, 3}
     assert m.roundtrip_std_set_int(FormalSetLike(1, 2, 3)) == {1, 2, 3}
diff --git a/tests/test_type_caster_pyobject_ptr.py b/tests/test_type_caster_pyobject_ptr.py
index 5df8ca0196..f9abd0063a 100644
--- a/tests/test_type_caster_pyobject_ptr.py
+++ b/tests/test_type_caster_pyobject_ptr.py
@@ -103,7 +103,8 @@ def test_return_list_pyobject_ptr_reference():
 
 def test_type_caster_name_via_incompatible_function_arguments_type_error():
     with pytest.raises(
-        TypeError, match=r"1\. \(arg0: object, arg1: typing.SupportsInt\) -> None"
+        TypeError,
+        match=r"1\. \(arg0: object, arg1: typing.SupportsInt \| typing.SupportsIndex\) -> None",
     ):
         m.pass_pyobject_ptr_and_int(ValueHolder(101), ValueHolder(202))