diff --git a/newsfragments/3185.fixed.md b/newsfragments/3185.fixed.md
new file mode 100644
index 00000000000..6b3a594c91a
--- /dev/null
+++ b/newsfragments/3185.fixed.md
@@ -0,0 +1 @@
+Fix conversion of classes implementing `__complex__` to `Complex` when using `abi3` or PyPy.
diff --git a/src/conversions/num_complex.rs b/src/conversions/num_complex.rs
index 217d862a542..df6b54b45bc 100644
--- a/src/conversions/num_complex.rs
+++ b/src/conversions/num_complex.rs
@@ -152,6 +152,18 @@ macro_rules! complex_conversion {
 
                 #[cfg(any(Py_LIMITED_API, PyPy))]
                 unsafe {
+                    let obj = if obj.is_instance_of::<PyComplex>() {
+                        obj
+                    } else if let Some(method) =
+                        obj.lookup_special(crate::intern!(obj.py(), "__complex__"))?
+                    {
+                        method.call0()?
+                    } else {
+                        // `obj` might still implement `__float__` or `__index__`, which will be
+                        // handled by `PyComplex_{Real,Imag}AsDouble`, including propagating any
+                        // errors if those methods don't exist / raise exceptions.
+                        obj
+                    };
                     let ptr = obj.as_ptr();
                     let real = ffi::PyComplex_RealAsDouble(ptr);
                     if real == -1.0 {
@@ -172,6 +184,7 @@ complex_conversion!(f64);
 #[cfg(test)]
 mod tests {
     use super::*;
+    use crate::types::PyModule;
 
     #[test]
     fn from_complex() {
@@ -197,4 +210,131 @@ mod tests {
             assert!(obj.extract::<Complex<f64>>(py).is_err());
         });
     }
+    #[test]
+    fn from_python_magic() {
+        Python::with_gil(|py| {
+            let module = PyModule::from_code(
+                py,
+                r#"
+class A:
+    def __complex__(self): return 3.0+1.2j
+class B:
+    def __float__(self): return 3.0
+class C:
+    def __index__(self): return 3
+                "#,
+                "test.py",
+                "test",
+            )
+            .unwrap();
+            let from_complex = module.getattr("A").unwrap().call0().unwrap();
+            assert_eq!(
+                from_complex.extract::<Complex<f64>>().unwrap(),
+                Complex::new(3.0, 1.2)
+            );
+            let from_float = module.getattr("B").unwrap().call0().unwrap();
+            assert_eq!(
+                from_float.extract::<Complex<f64>>().unwrap(),
+                Complex::new(3.0, 0.0)
+            );
+            // Before Python 3.8, `__index__` wasn't tried by `float`/`complex`.
+            #[cfg(Py_3_8)]
+            {
+                let from_index = module.getattr("C").unwrap().call0().unwrap();
+                assert_eq!(
+                    from_index.extract::<Complex<f64>>().unwrap(),
+                    Complex::new(3.0, 0.0)
+                );
+            }
+        })
+    }
+    #[test]
+    fn from_python_inherited_magic() {
+        Python::with_gil(|py| {
+            let module = PyModule::from_code(
+                py,
+                r#"
+class First: pass
+class ComplexMixin:
+    def __complex__(self): return 3.0+1.2j
+class FloatMixin:
+    def __float__(self): return 3.0
+class IndexMixin:
+    def __index__(self): return 3
+class A(First, ComplexMixin): pass
+class B(First, FloatMixin): pass
+class C(First, IndexMixin): pass
+                "#,
+                "test.py",
+                "test",
+            )
+            .unwrap();
+            let from_complex = module.getattr("A").unwrap().call0().unwrap();
+            assert_eq!(
+                from_complex.extract::<Complex<f64>>().unwrap(),
+                Complex::new(3.0, 1.2)
+            );
+            let from_float = module.getattr("B").unwrap().call0().unwrap();
+            assert_eq!(
+                from_float.extract::<Complex<f64>>().unwrap(),
+                Complex::new(3.0, 0.0)
+            );
+            #[cfg(Py_3_8)]
+            {
+                let from_index = module.getattr("C").unwrap().call0().unwrap();
+                assert_eq!(
+                    from_index.extract::<Complex<f64>>().unwrap(),
+                    Complex::new(3.0, 0.0)
+                );
+            }
+        })
+    }
+    #[test]
+    fn from_python_noncallable_descriptor_magic() {
+        // Functions and lambdas implement the descriptor protocol in a way that makes
+        // `type(inst).attr(inst)` equivalent to `inst.attr()` for methods, but this isn't the only
+        // way the descriptor protocol might be implemented.
+        Python::with_gil(|py| {
+            let module = PyModule::from_code(
+                py,
+                r#"
+class A:
+    @property
+    def __complex__(self):
+        return lambda: 3.0+1.2j
+                "#,
+                "test.py",
+                "test",
+            )
+            .unwrap();
+            let obj = module.getattr("A").unwrap().call0().unwrap();
+            assert_eq!(
+                obj.extract::<Complex<f64>>().unwrap(),
+                Complex::new(3.0, 1.2)
+            );
+        })
+    }
+    #[test]
+    fn from_python_nondescriptor_magic() {
+        // Magic methods don't need to implement the descriptor protocol, if they're callable.
+        Python::with_gil(|py| {
+            let module = PyModule::from_code(
+                py,
+                r#"
+class MyComplex:
+    def __call__(self): return 3.0+1.2j
+class A:
+    __complex__ = MyComplex()
+                "#,
+                "test.py",
+                "test",
+            )
+            .unwrap();
+            let obj = module.getattr("A").unwrap().call0().unwrap();
+            assert_eq!(
+                obj.extract::<Complex<f64>>().unwrap(),
+                Complex::new(3.0, 1.2)
+            );
+        })
+    }
 }
diff --git a/src/types/any.rs b/src/types/any.rs
index afdeb6ab573..d6ad2cbc4d7 100644
--- a/src/types/any.rs
+++ b/src/types/any.rs
@@ -124,6 +124,55 @@ impl PyAny {
         }
     }
 
+    /// Retrieve an attribute value, skipping the instance dictionary during the lookup but still
+    /// binding the object to the instance.
+    ///
+    /// This is useful when trying to resolve Python's "magic" methods like `__getitem__`, which
+    /// are looked up starting from the type object.  This returns an `Option` as it is not
+    /// typically a direct error for the special lookup to fail, as magic methods are optional in
+    /// many situations in which they might be called.
+    ///
+    /// To avoid repeated temporary allocations of Python strings, the [`intern!`] macro can be used
+    /// to intern `attr_name`.
+    #[allow(dead_code)] // Currently only used with num-complex+abi3, so dead without that.
+    pub(crate) fn lookup_special<N>(&self, attr_name: N) -> PyResult<Option<&PyAny>>
+    where
+        N: IntoPy<Py<PyString>>,
+    {
+        let py = self.py();
+        let self_type = self.get_type();
+        let attr = if let Ok(attr) = self_type.getattr(attr_name) {
+            attr
+        } else {
+            return Ok(None);
+        };
+
+        // Manually resolve descriptor protocol.
+        unsafe {
+            if cfg!(Py_3_10)
+                || ffi::PyType_HasFeature(attr.get_type_ptr(), ffi::Py_TPFLAGS_HEAPTYPE) != 0
+            {
+                // This is the preferred faster path, but does not work on static types (generally,
+                // types defined in extension modules) before Python 3.10.
+                let descr_get_ptr = ffi::PyType_GetSlot(attr.get_type_ptr(), ffi::Py_tp_descr_get);
+                if descr_get_ptr.is_null() {
+                    return Ok(Some(attr));
+                }
+                let descr_get: ffi::descrgetfunc = std::mem::transmute(descr_get_ptr);
+                let ret = descr_get(attr.as_ptr(), self.as_ptr(), self_type.as_ptr());
+                if ret.is_null() {
+                    Err(PyErr::fetch(py))
+                } else {
+                    Ok(Some(py.from_owned_ptr(ret)))
+                }
+            } else if let Ok(descr_get) = attr.get_type().getattr(crate::intern!(py, "__get__")) {
+                descr_get.call1((attr, self, self_type)).map(Some)
+            } else {
+                Ok(Some(attr))
+            }
+        }
+    }
+
     /// Sets an attribute value.
     ///
     /// This is equivalent to the Python expression `self.attr_name = value`.
@@ -974,9 +1023,82 @@ impl PyAny {
 #[cfg(test)]
 mod tests {
     use crate::{
-        types::{IntoPyDict, PyBool, PyList, PyLong, PyModule},
+        types::{IntoPyDict, PyAny, PyBool, PyList, PyLong, PyModule},
         Python, ToPyObject,
     };
+
+    #[test]
+    fn test_lookup_special() {
+        Python::with_gil(|py| {
+            let module = PyModule::from_code(
+                py,
+                r#"
+class CustomCallable:
+    def __call__(self):
+        return 1
+
+class SimpleInt:
+    def __int__(self):
+        return 1
+
+class InheritedInt(SimpleInt): pass
+
+class NoInt: pass
+
+class NoDescriptorInt:
+    __int__ = CustomCallable()
+
+class InstanceOverrideInt:
+    def __int__(self):
+        return 1
+instance_override = InstanceOverrideInt()
+instance_override.__int__ = lambda self: 2
+
+class ErrorInDescriptorInt:
+    @property
+    def __int__(self):
+        raise ValueError("uh-oh!")
+
+class NonHeapNonDescriptorInt:
+    # A static-typed callable that doesn't implement `__get__`.  These are pretty hard to come by.
+    __int__ = int
+                "#,
+                "test.py",
+                "test",
+            )
+            .unwrap();
+
+            let int = crate::intern!(py, "__int__");
+            let eval_int =
+                |obj: &PyAny| obj.lookup_special(int)?.unwrap().call0()?.extract::<u32>();
+
+            let simple = module.getattr("SimpleInt").unwrap().call0().unwrap();
+            assert_eq!(eval_int(simple).unwrap(), 1);
+            let inherited = module.getattr("InheritedInt").unwrap().call0().unwrap();
+            assert_eq!(eval_int(inherited).unwrap(), 1);
+            let no_descriptor = module.getattr("NoDescriptorInt").unwrap().call0().unwrap();
+            assert_eq!(eval_int(no_descriptor).unwrap(), 1);
+            let missing = module.getattr("NoInt").unwrap().call0().unwrap();
+            assert!(missing.lookup_special(int).unwrap().is_none());
+            // Note the instance override should _not_ call the instance method that returns 2,
+            // because that's not how special lookups are meant to work.
+            let instance_override = module.getattr("instance_override").unwrap();
+            assert_eq!(eval_int(instance_override).unwrap(), 1);
+            let descriptor_error = module
+                .getattr("ErrorInDescriptorInt")
+                .unwrap()
+                .call0()
+                .unwrap();
+            assert!(descriptor_error.lookup_special(int).is_err());
+            let nonheap_nondescriptor = module
+                .getattr("NonHeapNonDescriptorInt")
+                .unwrap()
+                .call0()
+                .unwrap();
+            assert_eq!(eval_int(nonheap_nondescriptor).unwrap(), 0);
+        })
+    }
+
     #[test]
     fn test_call_for_non_existing_method() {
         Python::with_gil(|py| {