diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index cd312b09ab6c1..c460e4fe23248 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -1356,21 +1356,20 @@ def median(self, axis: Optional[int] = None, skipna: bool = True, *args, **kwarg
         if axis is not None and abs(axis) >= self.ndim:
             raise ValueError("abs(axis) must be less than ndim")
 
-        if self.size == 0:
-            if self.ndim == 1 or axis is None:
-                return NaT
-            shape = list(self.shape)
-            del shape[axis]
-            shape = [1 if x == 0 else x for x in shape]
-            result = np.empty(shape, dtype="i8")
-            result.fill(iNaT)
+        if is_period_dtype(self.dtype):
+            # pass datetime64 values to nanops to get correct NaT semantics
+            result = nanops.nanmedian(
+                self._ndarray.view("M8[ns]"), axis=axis, skipna=skipna
+            )
+            result = result.view("i8")
+            if axis is None or self.ndim == 1:
+                return self._box_func(result)
             return self._from_backing_data(result)
 
-        mask = self.isna()
-        result = nanops.nanmedian(self.asi8, axis=axis, skipna=skipna, mask=mask)
+        result = nanops.nanmedian(self._ndarray, axis=axis, skipna=skipna)
         if axis is None or self.ndim == 1:
             return self._box_func(result)
-        return self._from_backing_data(result.astype("i8"))
+        return self._from_backing_data(result)
 
 
 class DatelikeOps(DatetimeLikeArrayMixin):
diff --git a/pandas/core/nanops.py b/pandas/core/nanops.py
index 46ff4a0e2f612..fdf27797db3ab 100644
--- a/pandas/core/nanops.py
+++ b/pandas/core/nanops.py
@@ -2,6 +2,7 @@
 import itertools
 import operator
 from typing import Any, Optional, Tuple, Union, cast
+import warnings
 
 import numpy as np
 
@@ -645,7 +646,11 @@ def get_median(x):
         mask = notna(x)
         if not skipna and not mask.all():
             return np.nan
-        return np.nanmedian(x[mask])
+        with warnings.catch_warnings():
+            # Suppress RuntimeWarning about All-NaN slice
+            warnings.filterwarnings("ignore", "All-NaN slice encountered")
+            res = np.nanmedian(x[mask])
+        return res
 
     values, mask, dtype, _, _ = _get_values(values, skipna, mask=mask)
     if not is_float_dtype(values.dtype):
@@ -673,7 +678,11 @@ def get_median(x):
                 )
 
             # fastpath for the skipna case
-            return _wrap_results(np.nanmedian(values, axis), dtype)
+            with warnings.catch_warnings():
+                # Suppress RuntimeWarning about All-NaN slice
+                warnings.filterwarnings("ignore", "All-NaN slice encountered")
+                res = np.nanmedian(values, axis)
+            return _wrap_results(res, dtype)
 
         # must return the correct shape, but median is not defined for the
         # empty set so return nans of shape "everything but the passed axis"
diff --git a/pandas/tests/arrays/test_datetimelike.py b/pandas/tests/arrays/test_datetimelike.py
index 463196eaa36bf..f621479e4f311 100644
--- a/pandas/tests/arrays/test_datetimelike.py
+++ b/pandas/tests/arrays/test_datetimelike.py
@@ -4,7 +4,7 @@
 import pytest
 import pytz
 
-from pandas._libs import OutOfBoundsDatetime, Timestamp
+from pandas._libs import NaT, OutOfBoundsDatetime, Timestamp
 from pandas.compat.numpy import np_version_under1p18
 
 import pandas as pd
@@ -456,6 +456,54 @@ def test_shift_fill_int_deprecated(self):
         expected[1:] = arr[:-1]
         tm.assert_equal(result, expected)
 
+    def test_median(self, arr1d):
+        arr = arr1d
+        if len(arr) % 2 == 0:
+            # make it easier to define `expected`
+            arr = arr[:-1]
+
+        expected = arr[len(arr) // 2]
+
+        result = arr.median()
+        assert type(result) is type(expected)
+        assert result == expected
+
+        arr[len(arr) // 2] = NaT
+        if not isinstance(expected, Period):
+            expected = arr[len(arr) // 2 - 1 : len(arr) // 2 + 2].mean()
+
+        assert arr.median(skipna=False) is NaT
+
+        result = arr.median()
+        assert type(result) is type(expected)
+        assert result == expected
+
+        assert arr[:0].median() is NaT
+        assert arr[:0].median(skipna=False) is NaT
+
+        # 2d Case
+        arr2 = arr.reshape(-1, 1)
+
+        result = arr2.median(axis=None)
+        assert type(result) is type(expected)
+        assert result == expected
+
+        assert arr2.median(axis=None, skipna=False) is NaT
+
+        result = arr2.median(axis=0)
+        expected2 = type(arr)._from_sequence([expected], dtype=arr.dtype)
+        tm.assert_equal(result, expected2)
+
+        result = arr2.median(axis=0, skipna=False)
+        expected2 = type(arr)._from_sequence([NaT], dtype=arr.dtype)
+        tm.assert_equal(result, expected2)
+
+        result = arr2.median(axis=1)
+        tm.assert_equal(result, arr)
+
+        result = arr2.median(axis=1, skipna=False)
+        tm.assert_equal(result, arr)
+
 
 class TestDatetimeArray(SharedTests):
     index_cls = pd.DatetimeIndex
@@ -465,7 +513,7 @@ class TestDatetimeArray(SharedTests):
     @pytest.fixture
     def arr1d(self, tz_naive_fixture, freqstr):
         tz = tz_naive_fixture
-        dti = pd.date_range("2016-01-01 01:01:00", periods=3, freq=freqstr, tz=tz)
+        dti = pd.date_range("2016-01-01 01:01:00", periods=5, freq=freqstr, tz=tz)
         dta = dti._data
         return dta
 
diff --git a/pandas/tests/arrays/test_datetimes.py b/pandas/tests/arrays/test_datetimes.py
index 9245eda2a71fe..66a92dd6f1cff 100644
--- a/pandas/tests/arrays/test_datetimes.py
+++ b/pandas/tests/arrays/test_datetimes.py
@@ -515,7 +515,7 @@ def test_median_empty(self, skipna, tz):
         tm.assert_equal(result, expected)
 
         result = arr.median(axis=1, skipna=skipna)
-        expected = type(arr)._from_sequence([pd.NaT], dtype=arr.dtype)
+        expected = type(arr)._from_sequence([], dtype=arr.dtype)
         tm.assert_equal(result, expected)
 
     def test_median(self, arr1d):