API/DEPR: interpolate with object dtype

[jbrockmendel]

import pandas as pd
import numpy as np

ser = pd.Series([1, np.nan, 2], dtype=object)

>>> ser.interpolate()
0      1
1    NaN
2      2
dtype: object

ATM if we have object-dtype, interpolate is incorrectly a no-op. (In the DataFrame case we have a weird check that raises on all-object).

If we disable the check in Block.interpolate if m is None and self.dtype.kind != "f": that short-circuits, then the OP case raises in np.interp when trying to cast to float64.

The viable options I see are:

1. Raise on object dtype, tell users to do .infer_objects() before calling interpolate.
2. Check if there the array has any NaNs. if not, short-circuit as in the status quo. Otherwise goto 1).

I lean towards "do 2) immediately and 1) with a deprecation cycle".

(I haven't checked the scipy paths, just the numpy one)

[topper-123]

Sounds good to me.

[verglasz]

I am currently using resample().interpolate() on a dataframe with a datetime index and which has some numeric columns and some columns with python strings, and the current (deprecated) behaviour is very convenient, I can fill the NaNs left in the string column with appropriate information based on the interpolation or other details or call some other method like bfill/ffill to take care of those.
Once this will error, what would be the approach to achieve the same result? Separate the numeric and object columns and call separately resample().interpolate() on the first set and resample().fillna() on the second, then put the df back together?
Imho it would be nice to keep some way of accessing the current behaviour (maybe some kind of allow_objects option?) so that the code can remain neater and not fiddle with individual columns (which is even more of a pain when writing utility function where the columns aren't known beforehand).