diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index c8fa6d46c3b7e..ed38c09129679 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -488,6 +488,7 @@ Groupby/resample/rolling
 - Bug in :meth:`DataFrame.rolling.corr` when the :class:`DataFrame` columns was a :class:`MultiIndex` (:issue:`21157`)
 - Bug in :meth:`DataFrame.groupby.rolling` when specifying ``on`` and calling ``__getitem__`` would subsequently return incorrect results (:issue:`43355`)
 - Bug in :meth:`GroupBy.apply` with time-based :class:`Grouper` objects incorrectly raising ``ValueError`` in corner cases where the grouping vector contains a ``NaT`` (:issue:`43500`, :issue:`43515`)
+- Bug in :meth:`GroupBy.mean` failing with ``complex`` dtype (:issue:`43701`)
 
 Reshaping
 ^^^^^^^^^
diff --git a/pandas/_libs/groupby.pyx b/pandas/_libs/groupby.pyx
index bbdc5a8287502..6dfed95e7afb6 100644
--- a/pandas/_libs/groupby.pyx
+++ b/pandas/_libs/groupby.pyx
@@ -481,6 +481,12 @@ ctypedef fused add_t:
     complex128_t
     object
 
+ctypedef fused mean_t:
+    float64_t
+    float32_t
+    complex64_t
+    complex128_t
+
 
 @cython.wraparound(False)
 @cython.boundscheck(False)
@@ -670,9 +676,9 @@ def group_var(floating[:, ::1] out,
 
 @cython.wraparound(False)
 @cython.boundscheck(False)
-def group_mean(floating[:, ::1] out,
+def group_mean(mean_t[:, ::1] out,
                int64_t[::1] counts,
-               ndarray[floating, ndim=2] values,
+               ndarray[mean_t, ndim=2] values,
                const intp_t[::1] labels,
                Py_ssize_t min_count=-1,
                bint is_datetimelike=False,
@@ -712,8 +718,8 @@ def group_mean(floating[:, ::1] out,
 
     cdef:
         Py_ssize_t i, j, N, K, lab, ncounts = len(counts)
-        floating val, count, y, t, nan_val
-        floating[:, ::1] sumx, compensation
+        mean_t val, count, y, t, nan_val
+        mean_t[:, ::1] sumx, compensation
         int64_t[:, ::1] nobs
         Py_ssize_t len_values = len(values), len_labels = len(labels)
 
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index 07353863cdc0b..89bf14d649a99 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -307,7 +307,7 @@ def _get_result_dtype(self, dtype: DtypeObj) -> DtypeObj:
         elif how in ["mean", "median", "var"]:
             if isinstance(dtype, (BooleanDtype, _IntegerDtype)):
                 return Float64Dtype()
-            elif is_float_dtype(dtype):
+            elif is_float_dtype(dtype) or is_complex_dtype(dtype):
                 return dtype
             elif is_numeric_dtype(dtype):
                 return np.dtype(np.float64)
diff --git a/pandas/tests/groupby/aggregate/test_aggregate.py b/pandas/tests/groupby/aggregate/test_aggregate.py
index 8d1962f7cab3b..2c798e543bf6b 100644
--- a/pandas/tests/groupby/aggregate/test_aggregate.py
+++ b/pandas/tests/groupby/aggregate/test_aggregate.py
@@ -1357,3 +1357,23 @@ def test_group_mean_datetime64_nat(input_data, expected_output):
 
     result = data.groupby([0, 0, 0]).mean()
     tm.assert_series_equal(result, expected)
+
+
+@pytest.mark.parametrize(
+    "func, output", [("mean", [8 + 18j, 10 + 22j]), ("sum", [40 + 90j, 50 + 110j])]
+)
+def test_groupby_complex(func, output):
+    # GH#43701
+    data = Series(np.arange(20).reshape(10, 2).dot([1, 2j]))
+    result = data.groupby(data.index % 2).agg(func)
+    expected = Series(output)
+    tm.assert_series_equal(result, expected)
+
+
+@pytest.mark.parametrize("func", ["min", "max", "var"])
+def test_groupby_complex_raises(func):
+    # GH#43701
+    data = Series(np.arange(20).reshape(10, 2).dot([1, 2j]))
+    msg = "No matching signature found"
+    with pytest.raises(TypeError, match=msg):
+        data.groupby(data.index % 2).agg(func)