BUG: series with complex nan
- [x] Closes #53627
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests)
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit)
- [x] Added an entry in the latest `doc/source/whatsnew/v2.1.0.rst` file

@jbrockmendel suggested changes made, thanks for your review!

One thing to notice is the repr:
```python
>>> import pandas as pd
>>> pd.Series([complex("nan")])
0   NaN
dtype: complex128
>>> pd.Series([1, complex("nan"), 2])
0    1.0+0.0j
1   N000a000N
2    2.0+0.0j
dtype: complex128
```
I don't think this is as expected, but I suppose this should be a separate issue? Thanks @Charlie-XIAO 

> I don't think this is as expected, but I suppose this should be a separate issue?

Yeah that appears to be a bug. Could you open a separate issue about that? Sure, will do soon.