diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 620823b9703ab..790895f5f4f81 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -464,7 +464,7 @@ def __new__(
         cls,
         data=None,
         dtype=None,
-        copy: bool = False,
+        copy: bool | None = None,
         name=None,
         tupleize_cols: bool = True,
     ) -> Index:
@@ -477,6 +477,12 @@ def __new__(
 
         data_dtype = getattr(data, "dtype", None)
 
+        if copy is None:
+            if isinstance(data, Index):
+                copy = False
+            else:
+                copy = True
+
         # range
         if isinstance(data, (range, RangeIndex)):
             result = RangeIndex(start=data, copy=copy, name=name)
@@ -6714,7 +6720,7 @@ def infer_objects(self, copy: bool = True) -> Index:
         )
         if copy and res_values is values:
             return self.copy()
-        return Index(res_values, name=self.name)
+        return Index(res_values, name=self.name, copy=False)
 
     # --------------------------------------------------------------------
     # Generated Arithmetic, Comparison, and Unary Methods
@@ -7051,7 +7057,7 @@ def ensure_index_from_sequences(sequences, names=None) -> Index:
         return MultiIndex.from_arrays(sequences, names=names)
 
 
-def ensure_index(index_like: Axes, copy: bool = False) -> Index:
+def ensure_index(index_like: Axes, copy: bool | None = None) -> Index:
     """
     Ensure that we have an index from some index-like object.
 
@@ -7059,7 +7065,8 @@ def ensure_index(index_like: Axes, copy: bool = False) -> Index:
     ----------
     index_like : sequence
         An Index or other sequence
-    copy : bool, default False
+    copy : bool, optional
+        By default copy if the passed object is not yet an Index object.
 
     Returns
     -------
@@ -7083,10 +7090,13 @@ def ensure_index(index_like: Axes, copy: bool = False) -> Index:
            )
     """
     if isinstance(index_like, Index):
-        if copy:
+        if copy is True:
             index_like = index_like.copy()
         return index_like
 
+    if copy is None:
+        copy = True
+
     if isinstance(index_like, ABCSeries):
         name = index_like.name
         return Index(index_like, name=name, copy=copy)
diff --git a/pandas/tests/copy_view/test_indexes.py b/pandas/tests/copy_view/test_indexes.py
new file mode 100644
index 0000000000000..c805b3f69f145
--- /dev/null
+++ b/pandas/tests/copy_view/test_indexes.py
@@ -0,0 +1,44 @@
+from pandas import (
+    DataFrame,
+    Index,
+    Series,
+)
+import pandas._testing as tm
+
+
+def test_index_constructor():
+    ser = Series([1, 2, 3])
+    idx = Index(ser)
+    # assert not np.shares_memory(idx.values, get_array(ser))
+    ser.iloc[0] = 0
+    tm.assert_index_equal(idx, Index([1, 2, 3]))
+
+
+def test_series_constructor():
+    ser = Series([1, 2, 3])
+    ser2 = Series(ser, index=ser)
+    ser2.iloc[0] = 0
+    tm.assert_index_equal(ser2.index, Index([1, 2, 3]))
+
+
+def test_dataframe_constructor():
+    ser = Series([1, 2, 3])
+    df = DataFrame({"a": ser}, index=ser)
+    ser.iloc[0] = 0
+    tm.assert_index_equal(df.index, Index([1, 2, 3]))
+    df.iloc[0, 0] = 10
+    tm.assert_index_equal(df.index, Index([1, 2, 3]))
+
+
+def test_dataframe_set_index_inplace():
+    df = DataFrame({"a": [1, 2, 3], "b": [4, 5, 6], "c": [0.1, 0.2, 0.3]})
+    df.set_index("a", drop=False, inplace=True)
+    df.iloc[0, 0] = 0
+    tm.assert_index_equal(df.index, Index([1, 2, 3], name="a"))
+
+
+def test_index_attribute_assignment():
+    ser = Series([1, 2, 3])
+    ser.index = ser
+    ser.iloc[0] = 10
+    tm.assert_index_equal(ser.index, Index([1, 2, 3]))
diff --git a/pandas/tests/reductions/test_stat_reductions.py b/pandas/tests/reductions/test_stat_reductions.py
index dd6aef04a2e6a..ce33f5279bdd0 100644
--- a/pandas/tests/reductions/test_stat_reductions.py
+++ b/pandas/tests/reductions/test_stat_reductions.py
@@ -77,6 +77,7 @@ def test_td64_mean(self, box):
         assert result == expected
 
         tdarr[0] = pd.NaT
+        obj = box(tdarr)
         assert obj.mean(skipna=False) is pd.NaT
 
         result2 = obj.mean(skipna=True)