diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index 388c5dbf6a7ee..2175a18a88e73 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -326,6 +326,7 @@ Numeric
 Conversion
 ^^^^^^^^^^
 - Bug in :meth:`Series.to_dict` with ``orient='records'`` now returns python native types (:issue:`25969`)
+- Bug in :meth:`Series.view` and :meth:`Index.view` when converting between datetime-like (``datetime64[ns]``, ``datetime64[ns, tz]``, ``timedelta64``, ``period``) dtypes (:issue:`39788`)
 -
 -
 
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index 5dd55ff0f1fa2..a6ee95e131559 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -44,6 +44,7 @@
 )
 from pandas._libs.tslibs.timestamps import integer_op_not_supported
 from pandas._typing import (
+    ArrayLike,
     DatetimeLikeScalar,
     Dtype,
     DtypeObj,
@@ -79,6 +80,10 @@
     is_unsigned_integer_dtype,
     pandas_dtype,
 )
+from pandas.core.dtypes.dtypes import (
+    DatetimeTZDtype,
+    PeriodDtype,
+)
 from pandas.core.dtypes.missing import (
     is_valid_na_for_dtype,
     isna,
@@ -428,9 +433,30 @@ def astype(self, dtype, copy=True):
         else:
             return np.asarray(self, dtype=dtype)
 
-    def view(self, dtype: Optional[Dtype] = None):
+    def view(self, dtype: Optional[Dtype] = None) -> ArrayLike:
+        # We handle datetime64, datetime64tz, timedelta64, and period
+        #  dtypes here. Everything else we pass through to the underlying
+        #  ndarray.
         if dtype is None or dtype is self.dtype:
             return type(self)(self._ndarray, dtype=self.dtype)
+
+        if isinstance(dtype, type):
+            # we sometimes pass non-dtype objects, e.g np.ndarray;
+            #  pass those through to the underlying ndarray
+            return self._ndarray.view(dtype)
+
+        dtype = pandas_dtype(dtype)
+        if isinstance(dtype, (PeriodDtype, DatetimeTZDtype)):
+            cls = dtype.construct_array_type()
+            return cls._simple_new(self.asi8, dtype=dtype)
+        elif dtype == "M8[ns]":
+            from pandas.core.arrays import DatetimeArray
+
+            return DatetimeArray._simple_new(self.asi8, dtype=dtype)
+        elif dtype == "m8[ns]":
+            from pandas.core.arrays import TimedeltaArray
+
+            return TimedeltaArray._simple_new(self.asi8.view("m8[ns]"), dtype=dtype)
         return self._ndarray.view(dtype=dtype)
 
     # ------------------------------------------------------------------
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 71095b8f4113a..7d8d964f63496 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -793,6 +793,23 @@ def view(self, cls=None):
         # we need to see if we are subclassing an
         # index type here
         if cls is not None and not hasattr(cls, "_typ"):
+            dtype = cls
+            if isinstance(cls, str):
+                dtype = pandas_dtype(cls)
+
+            if isinstance(dtype, (np.dtype, ExtensionDtype)) and needs_i8_conversion(
+                dtype
+            ):
+                if dtype.kind == "m" and dtype != "m8[ns]":
+                    # e.g. m8[s]
+                    return self._data.view(cls)
+
+                arr = self._data.view("i8")
+                idx_cls = self._dtype_to_subclass(dtype)
+                arr_cls = idx_cls._data_cls
+                arr = arr_cls._simple_new(self._data.view("i8"), dtype=dtype)
+                return idx_cls._simple_new(arr, name=self.name)
+
             result = self._data.view(cls)
         else:
             result = self._view()
diff --git a/pandas/tests/series/methods/test_view.py b/pandas/tests/series/methods/test_view.py
index 48f7b47f6d25a..f0069cdb9b79c 100644
--- a/pandas/tests/series/methods/test_view.py
+++ b/pandas/tests/series/methods/test_view.py
@@ -1,5 +1,10 @@
+import numpy as np
+import pytest
+
 from pandas import (
+    Index,
     Series,
+    array as pd_array,
     date_range,
 )
 import pandas._testing as tm
@@ -19,3 +24,23 @@ def test_view_tz(self):
             ]
         )
         tm.assert_series_equal(result, expected)
+
+    @pytest.mark.parametrize(
+        "first", ["m8[ns]", "M8[ns]", "M8[ns, US/Central]", "period[D]"]
+    )
+    @pytest.mark.parametrize(
+        "second", ["m8[ns]", "M8[ns]", "M8[ns, US/Central]", "period[D]"]
+    )
+    @pytest.mark.parametrize("box", [Series, Index, pd_array])
+    def test_view_between_datetimelike(self, first, second, box):
+
+        dti = date_range("2016-01-01", periods=3)
+
+        orig = box(dti)
+        obj = orig.view(first)
+        assert obj.dtype == first
+        tm.assert_numpy_array_equal(np.asarray(obj.view("i8")), dti.asi8)
+
+        res = obj.view(second)
+        assert res.dtype == second
+        tm.assert_numpy_array_equal(np.asarray(obj.view("i8")), dti.asi8)