BUG: complex Series/DataFrame display all complex nans as nan+0j

[Charlie-XIAO]

Pandas version checks

• I have checked that this issue has not already been reported.
• This bug does not exist on the latest version of pandas because complex nans in Series or DataFrame will raise in that version.
• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

>>> import pandas as pd
>>> c1 = complex(float("nan"), 2)
>>> c2 = complex(2, float("nan"))
>>> c3 = complex(float("nan"), float("nan"))
>>> pd.Series([1, c1, 2])
0    1.0+0.0j
1    NaN+0.0j
2    2.0+0.0j
dtype: complex128
>>> pd.Series([1, c2, 2]) 
0    1.0+0.0j
1    NaN+0.0j
2    2.0+0.0j
dtype: complex128
>>> pd.Series([1, c3, 2]) 
0    1.0+0.0j
1    NaN+0.0j
2    2.0+0.0j
dtype: complex128

Issue Description

Related to #53682 and #53762. When dealing with those two issues, I incorrectly assumed that complex nans are all in the form nan+0j, but in fact it is considered nan (i.e., isnan evaluates to True) if either the real or imaginary part is nan, for instance nan+2j, 2+nanj, nan+nanj, etc. IMO we need more explicit rules than isnan when distinguishing different kinds of complex nans, and maybe the display part need to be adjusted as well since previously I only considered the case that the real part can be nan. I will do further investigations and try to make a PR ASAP.

Note that this is completely a display issue because ser.array stores the values correctly.

Expected Behavior

>>> import pandas as pd
>>> c1 = complex(float("nan"), 2)
>>> c2 = complex(2, float("nan"))
>>> c3 = complex(float("nan"), float("nan"))
>>> pd.Series([1, c1, 2])
0    1.0+0.0j
1    NaN+2.0j
2    2.0+0.0j
dtype: complex128
>>> pd.Series([1, c2, 2]) 
0    1.0+0.0j
1    2.0+NaNj
2    2.0+0.0j
dtype: complex128
>>> pd.Series([1, c3, 2]) 
0    1.0+0.0j
1    NaN+NaNj
2    2.0+0.0j
dtype: complex128

Installed Versions

Installed versions

commit           : 54bf475fd4d38a08a353a47e44dfecce24cdfb4b
python           : 3.9.6.final.0
python-bits      : 64
OS               : Windows
OS-release       : 10
Version          : 10.0.22621
machine          : AMD64
processor        : Intel64 Family 6 Model 158 Stepping 10, GenuineIntel
byteorder        : little
LC_ALL           : None
LANG             : en_US.UTF-8
LOCALE           : Chinese (Simplified)_China.936

pandas           : 2.1.0.dev0+1045.g54bf475fd4
numpy            : 1.24.3
pytz             : 2023.3
dateutil         : 2.8.2
setuptools       : 56.0.0
pip              : 21.1.3
Cython           : 0.29.33
pytest           : 7.3.2
hypothesis       : 6.78.2
sphinx           : 6.2.1
blosc            : 1.11.1
feather          : None
xlsxwriter       : 3.1.2
lxml.etree       : 4.9.2
html5lib         : 1.1
pymysql          : 1.0.3
psycopg2         : 2.9.6
jinja2           : 3.1.2
IPython          : 8.14.0
pandas_datareader: None
bs4              : 4.12.2
bottleneck       : 1.3.7
brotli           :
fastparquet      : 2023.4.0
fsspec           : 2023.6.0
gcsfs            : 2023.6.0
matplotlib       : 3.7.1
numba            : 0.57.0
numexpr          : 2.8.4
odfpy            : None
openpyxl         : 3.1.2
pandas_gbq       : None
pyarrow          : 12.0.1
pyreadstat       : 1.2.2
pyxlsb           : 1.0.10
s3fs             : 2023.6.0
scipy            : 1.10.1
snappy           :
sqlalchemy       : 2.0.16
tables           : 3.8.0
tabulate         : 0.9.0
xarray           : 2023.5.0
xlrd             : 2.0.1
zstandard        : 0.21.0
tzdata           : 2023.3
qtpy             : None
pyqt5            : None