diff --git a/pandas-stubs/core/indexes/base.pyi b/pandas-stubs/core/indexes/base.pyi
index 1866e9464..87b4a1d0f 100644
--- a/pandas-stubs/core/indexes/base.pyi
+++ b/pandas-stubs/core/indexes/base.pyi
@@ -3,7 +3,7 @@ from collections.abc import (
     Callable,
     Hashable,
     Iterable,
-    MutableMapping,
+    Mapping,
     Sequence,
 )
 from datetime import (
@@ -104,6 +104,7 @@ from pandas._typing import (
     SequenceNotStr,
     SliceType,
     SupportsDType,
+    TakeIndexer,
     TimedeltaDtypeArg,
     TimestampDtypeArg,
     np_1darray,
@@ -350,40 +351,37 @@ class Index(IndexOpsMixin[S1], ElementOpsMixin[S1]):
         Index,
     ]: ...
     @final
-    def is_(self, other) -> bool: ...
+    def is_(self, other: object) -> bool: ...
     def __len__(self) -> int: ...
     def __array__(
         self, dtype: _str | np.dtype = ..., copy: bool | None = ...
     ) -> np_1darray: ...
-    def __array_wrap__(self, result, context=...): ...
     @property
     def dtype(self) -> DtypeObj: ...
     @final
-    def ravel(self, order: _str = ...): ...
+    def ravel(self, order: _str = "C") -> Self: ...
     def view(self, cls=...): ...
     def astype(self, dtype: DtypeArg, copy: bool = True) -> Index: ...
     def take(
         self,
-        indices,
+        indices: TakeIndexer,
         axis: int = 0,
         allow_fill: bool = True,
         fill_value: Scalar | None = None,
         **kwargs: Any,
+    ) -> Self: ...
+    def repeat(
+        self, repeats: int | AnyArrayLikeInt | Sequence[int], axis: None = None
     ): ...
-    def repeat(self, repeats, axis=...): ...
     def copy(self, name: Hashable = ..., deep: bool = False) -> Self: ...
-    @final
-    def __copy__(self, **kwargs: Any): ...
-    @final
-    def __deepcopy__(self, memo: MutableMapping[int, Any] | None = None) -> Self: ...
     def format(
         self, name: bool = ..., formatter: Callable | None = ..., na_rep: _str = ...
     ) -> list[_str]: ...
-    def to_flat_index(self): ...
+    def to_flat_index(self) -> Index: ...
     def to_series(
         self, index: Index | None = None, name: Hashable | None = None
     ) -> Series[S1]: ...
-    def to_frame(self, index: bool = True, name=...) -> DataFrame: ...
+    def to_frame(self, index: bool = True, name: Hashable = ...) -> DataFrame: ...
     @property
     def name(self) -> Hashable | None: ...
     @name.setter
@@ -392,11 +390,17 @@ class Index(IndexOpsMixin[S1], ElementOpsMixin[S1]):
     def names(self) -> list[Hashable | None]: ...
     @names.setter
     def names(self, names: SequenceNotStr[Hashable | None]) -> None: ...
-    def set_names(self, names, *, level=..., inplace: bool = ...): ...
+    def set_names(
+        self,
+        names: Hashable | Sequence[Hashable] | Mapping[Any, Hashable],
+        *,
+        level: Level | Sequence[Level] | None = None,
+        inplace: bool = False,
+    ) -> Self: ...
     @overload
-    def rename(self, name, *, inplace: Literal[False] = False) -> Self: ...
+    def rename(self, name: Hashable, *, inplace: Literal[False] = False) -> Self: ...
     @overload
-    def rename(self, name, *, inplace: Literal[True]) -> None: ...
+    def rename(self, name: Hashable, *, inplace: Literal[True]) -> None: ...
     @property
     def nlevels(self) -> int: ...
     def get_level_values(self, level: int | _str) -> Index: ...
diff --git a/pandas-stubs/core/indexes/multi.pyi b/pandas-stubs/core/indexes/multi.pyi
index d44b42bd6..a7076ade4 100644
--- a/pandas-stubs/core/indexes/multi.pyi
+++ b/pandas-stubs/core/indexes/multi.pyi
@@ -109,7 +109,7 @@ class MultiIndex(Index):
     def dropna(self, how: AnyAll = "any") -> Self: ...
     def get_level_values(self, level: str | int) -> Index: ...
     def unique(self, level=...): ...
-    def to_frame(
+    def to_frame(  # type: ignore[override]
         self,
         index: bool = True,
         name: list[HashableT] = ...,
diff --git a/tests/indexes/test_indexes.py b/tests/indexes/test_indexes.py
index 8647f6dde..f75e86fd9 100644
--- a/tests/indexes/test_indexes.py
+++ b/tests/indexes/test_indexes.py
@@ -1523,3 +1523,16 @@ def test_datetimeindex_where() -> None:
 
     val_range = pd.RangeIndex(2).where(pd.Series([True, False]), 3)
     check(assert_type(val_range, pd.Index), pd.RangeIndex)
+
+
+def test_index_set_names() -> None:
+    """Test Index.where with multiple types of other GH1419."""
+    idx = pd.Index([1, 2])
+    check(assert_type(idx.set_names("chinchilla"), "pd.Index[int]"), pd.Index, np.int64)
+    check(
+        assert_type(idx.set_names(["chinchilla"]), "pd.Index[int]"), pd.Index, np.int64
+    )
+
+    mi = pd.MultiIndex.from_arrays([[1, 2, 3], [4, 5, 6]], names=["elk", "owl"])
+    mi.set_names(["beluga", "pig"])
+    mi.set_names({"elk": "beluga", "owl": "pig"})