REF: Replace np.find_common_dtype with np.result_type
In preparation of https://github.com/numpy/numpy/pull/22539

Hmmmm, the fact that `find_common_type` bails on datetimes (if there is more than one) and returns object dtype when it doesn't know what to do seems to confuse things?  Sounds like I should document the fact that `find_common_type` falls back to `object` and that things may be a bit strange if datetime/timedelta are involved. A philosophical question: does `np.result_type` provide the type after applying an _operation_ (like arithmetic) to multiple different types? I think in pandas, `np.find_common_dtype` is used to answer "what dtype can _store_ multiple different types" >  Sounds like I should document the fact that find_common_type falls back to object and that things may be a bit strange if datetime/timedelta are involved.

Yeah I think pandas appreciates that `datetime/timedelta` with `non-datetime/timedelta` goes to `object` in `find_common_dtype`. I notice `result_type` will raise a `TypeError`, but I guess pandas can catch and know it should be object

But I think `np.result_type(datetime, timedelta)` returning `dtype('<M8[ns]')` is odd for pandas given the "storage" comment I made above.

```
In [13]: np.__version__
Out[13]: '1.23.1'

----

In [14]: np.find_common_type((np.dtype("datetime64[ns]"), np.dtype("timedelta64[ns]")), [])
Out[14]: dtype('O')  # from a "storage" perspective, this makes sense

In [15]: np.result_type(*(np.dtype("datetime64[ns]"), np.dtype("timedelta64[ns]")))
Out[15]: dtype('<M8[ns]')  # from datetime +/- timedelta this makes sense

----

In [17]: np.find_common_type((np.dtype("datetime64[ns]"), np.int64), [])
Out[17]: dtype('O')


# I think this TypeError below can be okay if it can always signal to pandas that "object" is needed to store these types
In [18]: np.result_type(*(np.dtype("datetime64[ns]"), np.int64))
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Input In [18], in <cell line: 1>()
----> 1 np.result_type(*(np.dtype("datetime64[ns]"), np.int64))

TypeError: The DTypes <class 'numpy.dtype[datetime64]'> and <class 'numpy.dtype[int64]'> do not have a common DType. For example they cannot be stored in a single array unless the dtype is `object`.

----
``` > is odd for pandas given the "storage" comment I made above

It is odd/wrong in NumPy as well.  Another problem is timedelta+integers.  I wanted to look into that a bit and think about deprecating it (this doesn't help you immediately maybe).

I suspect the reason is to make certain math promotions work easier (i.e. find timedelta + int as a loop that uses timedelta+timedelta), and there should be better solutions for that.

EDIT: I can't really think of an obvious workaround right now.  Unless there is only one input, I think a non-numeric dtype return was effectively always object, so:
```
try:
    common_dtype = np.result_type(*dtypes)
    if common_dtype.kind in "mMSU":
        # NumPy promotion currently misbehaves for for times and strings,
        # so fall back to object (find_common_dtype did unless there was only one dtype)
        common_dtype = np.dtype('O')
except TypeError:
    # Fall back to object on error (could be more specific error eventually)
    common_dtype = np.dtype('O')
```
might work, but feels also a bit clunky.  (The above would allow certain structured dtypes to promote correctly for example).  So there might be a point in not deprecating and first cleaning up the bad time promotion (and strings). >  Another problem is timedelta+integers.

From a pandas point of view, `object` would be the correct "result type"  in terms of storing timedelta and ints. I think generally pandas would support `np.result_type(datetime/timedelta, anything else)` to be a `TypeError` or `object`

> might work, but feels also a bit clunky.

Yeah we would probably migrate `np.find_common_dtype` to that code snippet if you went forward with the deprecation. It's good to see that `result_type` mostly matches `find_common_dtype though. Yeah, looking into fixing the timedelta/datetime promotion rules (i.e. its nonsense that they promote with bool/integers or with each other).
But it is probably a bit bigger and comes with some other fallout (as typical).  That might clear up enough, but I am not sure yet. This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. Closing this for now since we can't fully use `np.result_type` everywhere yet