This is more than infuriating. My rage level is really high over this. I've followed all the suggestions listed here, and the only one that PARTIALLY worked was deleting my custom email account and adding it again. I say "partially" because I still don't receive notifications if the app is closed, only if it's minimized.
What's worse? I've uninstalled the new Outlook no less than SEVEN TIMES, and Microsoft continues to reinstall this stupid app WITHOUT ASKING ME and forces it to open when I open the old mail app.
If you're going to force us all to use your new junk, which also forces us to get ADS in our inboxes when your old app didn't do that, then FIX YOUR JUNK.