ComputerWorld: Mailplane works perfectly with “Priority Inbox”
And here's what he wrote about Mailplane:
Similarly, Priority Inbox breaks new mail notifications on desktop mail as well. Fortunately, for Mac users, there's a workaround: Mailplane is a great $24.95 Gmail client for the desktop, I've been using it for a long time. The developers are working on a new version, still in beta, which allows you to customize desktop notifications to only show new Priority Inbox messages, or new messages with any user-configurable label. I've been using a it all week, it works perfectly.
If you like to try the brand new "Priority Inbox" in Mailplane, visit this download page. Make sure to read the release notes found on the same page.
Read the full blog post: Gmail Priority Inbox, 10 days later