I'm not talking about changing the favicon/page title, but actually hooking into Chrome's notification system. I just experienced this on a Discourse forum, and it was really slick. I think Firefox and Safari also support this, not sure about IE.
I expect this will eventually land in core, but it might be quite a while, so hoping an addon supports this now.
To keep the signal to noise ratio high, I'd only want it enabled for message replies and new PMs... ie, activity where you'd be likely to immediately respond. Probably not for likes (optional feature).
But on one of my Xenforo forums, I think this functionality would spike engagement because users would start responding faster to each other.