DataArray.idxmax converts coordinates into float64 by default

What happened?

Same example as in DataArray.idxmax but instead we look at the "y" dimension.
The starting "y" coordinates are of type int: [-1,0,1]

The return values of argmax are of type int64: good.
The return values of idxmax are of type float64: bad.

What did you expect to happen?

If no fillna operation must occur, then the return values of idxmax should be the same type as from the input.

Else, the return type might change to a new type depending on the type of the filled value.

Minimal Complete Verifiable Example

array = xr.DataArray(
    [
        [2.0, 1.0, 2.0, 0.0, -2.0],
        [-4.0, np.NaN, 2.0, np.NaN, -2.0],
        [np.NaN, np.NaN, 1.0, np.NaN, np.NaN],
    ],
    dims=["y", "x"],
    coords={"y": [-1, 0, 1], "x": np.arange(5.0) ** 2},
)
print(array.argmax(dim="y").dtype)
print(array.idxmax(dim="y").dtype)

MVCE confirmation

• Minimal example — the example is as focused as reasonably possible to demonstrate the underlying issue in xarray.
• Complete example — the example is self-contained, including all data and the text of any traceback.
• Verifiable example — the example copy & pastes into an IPython prompt or Binder notebook, returning the result.
• New issue — a search of GitHub Issues suggests this is not a duplicate.

Relevant log output

In [41]: print(array.argmax(dim="y").dtype)
int64

In [42]: print(array.idxmax(dim="y").dtype)
float64

Anything else we need to know?

Suggestions:

• change these two lines:

    if skipna or (skipna is None and array.dtype.kind in na_dtypes):
        # Put the NaN values back in after removing them

into

    if (skipna or (skipna is None and array.dtype.kind in na_dtypes)) and allna.any():
        # Put the NaN values back in after removing them, if any

• or maybe instead, it is a bug from DataArray.where: this res = res.where(~allna, fill_value) should not change the array type if not allna.any()? Actually, it is a known limitation of where: #3570

Environment

The idxmax handling is in xarray/core/computation.py around lines 2086-2088; begin by running the provided DataArray example and inspecting the allna/where path. Add regression coverage for integer y coordinates, then verify idxmax preserves the input coordinate dtype when no fill value is needed.