As discussed in #12658, we'll want to target NaT argmin, argmax, min, and max behavior adjustments to match the bug fix in sorting described there (to match NaN behavior) with 1.17 release target.

In particular, @seberg notes (comment)

It seems currently NaT is completely ignored for min/max and argmin/argmax. I wonder if we should push to change that to be identical NaN (NaN is always returned). I guess if it that bites someone, on the up-side it probably will be loud since it points to NaT.

There is also an independend bug for argmin when it starts with NaT and has only one not NaT item at the end