diff --git a/doc/source/whatsnew/v0.23.1.txt b/doc/source/whatsnew/v0.23.1.txt
index 9382d74f95295..c8a2076064c02 100644
--- a/doc/source/whatsnew/v0.23.1.txt
+++ b/doc/source/whatsnew/v0.23.1.txt
@@ -44,6 +44,8 @@ Documentation Changes
 Bug Fixes
 ~~~~~~~~~
 
+- tab completion on :class:`Index` in IPython no longer outputs deprecation warnings (:issue:`21125`)
+
 Groupby/Resample/Rolling
 ^^^^^^^^^^^^^^^^^^^^^^^^
 
diff --git a/pandas/core/accessor.py b/pandas/core/accessor.py
index c638b9e4ea117..7a853d575aa69 100644
--- a/pandas/core/accessor.py
+++ b/pandas/core/accessor.py
@@ -12,7 +12,8 @@
 
 class DirNamesMixin(object):
     _accessors = frozenset([])
-    _deprecations = frozenset(['asobject'])
+    _deprecations = frozenset(
+        ['asobject', 'base', 'data', 'flags', 'itemsize', 'strides'])
 
     def _dir_deletions(self):
         """ delete unwanted __dir__ for this object """
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index f4fa547574b9e..1e4dd2921b3f5 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -2088,6 +2088,17 @@ def test_get_duplicates_deprecated(self):
         with tm.assert_produces_warning(FutureWarning):
             index.get_duplicates()
 
+    def test_tab_complete_warning(self, ip):
+        # https://github.com/pandas-dev/pandas/issues/16409
+        pytest.importorskip('IPython', minversion="6.0.0")
+        from IPython.core.completer import provisionalcompleter
+
+        code = "import pandas as pd; idx = pd.Index([1, 2])"
+        ip.run_code(code)
+        with tm.assert_produces_warning(None):
+            with provisionalcompleter('ignore'):
+                list(ip.Completer.completions('idx.', 4))
+
 
 class TestMixedIntIndex(Base):
     # Mostly the tests from common.py for which the results differ