diff --git a/pandas/_libs/tslibs/timestamps.pyx b/pandas/_libs/tslibs/timestamps.pyx
index ec8478f3fc82a..afa678a6751af 100644
--- a/pandas/_libs/tslibs/timestamps.pyx
+++ b/pandas/_libs/tslibs/timestamps.pyx
@@ -856,12 +856,11 @@ cdef class _Timestamp(ABCTimestamp):
             local_val = self._maybe_convert_value_to_local()
             int64_t normalized
             int64_t ppd = periods_per_day(self._reso)
-
-        if self._reso != NPY_FR_ns:
-            raise NotImplementedError(self._reso)
+            _Timestamp ts
 
         normalized = normalize_i8_stamp(local_val, ppd)
-        return Timestamp(normalized).tz_localize(self.tzinfo)
+        ts = type(self)._from_value_and_reso(normalized, reso=self._reso, tz=None)
+        return ts.tz_localize(self.tzinfo)
 
     # -----------------------------------------------------------------
     # Pickle Methods
@@ -2035,6 +2034,8 @@ default 'raise'
         NaT
         """
         if self._reso != NPY_FR_ns:
+            if tz is None and self.tz is None:
+                return self
             raise NotImplementedError(self._reso)
 
         if ambiguous == 'infer':
diff --git a/pandas/tests/scalar/timestamp/test_timestamp.py b/pandas/tests/scalar/timestamp/test_timestamp.py
index 292c8f95ec2e6..36c3803ef41ff 100644
--- a/pandas/tests/scalar/timestamp/test_timestamp.py
+++ b/pandas/tests/scalar/timestamp/test_timestamp.py
@@ -715,11 +715,11 @@ def test_non_nano_construction(self, dt64, ts, reso):
         assert ts.value == dt64.view("i8")
 
         if reso == "s":
-            assert ts._reso == 7
+            assert ts._reso == NpyDatetimeUnit.NPY_FR_s.value
         elif reso == "ms":
-            assert ts._reso == 8
+            assert ts._reso == NpyDatetimeUnit.NPY_FR_ms.value
         elif reso == "us":
-            assert ts._reso == 9
+            assert ts._reso == NpyDatetimeUnit.NPY_FR_us.value
 
     def test_non_nano_fields(self, dt64, ts):
         alt = Timestamp(dt64)
@@ -830,6 +830,12 @@ def test_pickle(self, ts):
         assert rt._reso == ts._reso
         assert rt == ts
 
+    def test_normalize(self, dt64, ts):
+        alt = Timestamp(dt64)
+        result = ts.normalize()
+        assert result._reso == ts._reso
+        assert result == alt.normalize()
+
     def test_asm8(self, dt64, ts):
         rt = ts.asm8
         assert rt == dt64