ENH: add NDArrayBackedExtensionArray to public API
- [ ] closes #xxxx (Replace xxxx with the GitHub issue number)
- [ ] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [ ] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [ ] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

Reviving https://github.com/pandas-dev/pandas/pull/45544 as I'd like to use it for an UncertaintyArray.
Can we make this "public" with some kind of documentation that we may adjust it in non-user-facing ways more aggressively than we do with regular APIs? > Can we make this "public" with some kind of documentation that we may adjust it in non-user-facing ways more aggressively than we do with regular APIs?

added a note to the class docstring I'm unsure what to do about this error as `to_numpy` is an inherited method:
```
Error: /home/runner/work/pandas/pandas/pandas/core/arrays/base.py:542:EX01:pandas.api.extensions.NDArrayBackedExtensionArray.to_numpy:No examples section found
```

Also not sure what to do about 
```
Error: /home/runner/work/pandas/pandas/pandas/core/arrays/datetimes.py:179:EX03:pandas.arrays.DatetimeArray:flake8 error: line 2, col 4: E121 continuation line under-indented for hanging indent
```
as I haven't changed that file. Given that the Pandas team was finalizing the 2.2 release, you may have hit the docs system at a bad time.  I've had that happen to me.  Might make sense to try merging with latest and resubmitting. I tried jumping in to see if I could fix the missing toctree problem, but two things defeated me:

1. In the normal pandas build process (running `doc/make.py --warnings-are-errors`) a child process segfaulted on Mac OS 14.1.2 in a Python 3.11.4 enviroment
2. When I tried running `doc/make.py --warnings-are-errors --single reference/api/pandas.api.extensions.NDArrayBackedExtensionArray.argmax.rst` (a command that failed with the toctree warning), that single file built successfully.

I'm trying to see if I can isolate a reproducer for (1) above. cc @jreback @jorisvandenbossche from original PR 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. Thanks for the pull request, but it appears to have gone stale. If interested in continuing, please merge in the main branch, address any review comments and/or failing tests, and we can reopen.