What is wrong with continuing to publish with my PhD adviser? I have heard that it is 'bad' to continue to publish with your adviser after you get your PhD. Or is this something that is debated in the academic community?

In my situation, after my PhD, I can get a job in industry, but I am allowed time (half a day a week) to continue my research at my local university. This allows me to continue researching something that I am interested in, rather than researching something that someone else is interested in.

I can understand that every publication should not be between myself and my adviser. And I can understand that I should show that I am an independent researcher. But does that mean I should stop publishing with my (ex-)adviser?
There's nothing “wrong” with continuing to publish with your (ex-)adviser. You have correctly identified the issue to avoid: after getting your PhD, the next step in your career is to establish yourself as an independent researcher. Thus, of course: depending on the context, publishing with your adviser might send the wrong signal!

So, you should continue to publish with her, but you should make sure that it's not 100% of your research (and publications). Another point that makes a big difference is if you can, maybe now or maybe after some time, take the lead in that collaboration.