Recently, there was a system in place that told me if there was/were question(s) with new activity. There was also an indication seen in the tab just like when there were new chat messages.
But, I am no more ssing such indications. Was that feature revoked by the SE team, or it is a bug.
BTW, if was revoked by the team, then, this is a feature request as I found it most useful.
Some users did not like that notifications system, for example (here) and it might not be a good idea to put it back in place. In such cases, I see a system that user can turn on and off at will, that could be put in place.
Let the votes decide if this or some other conducive system to be put in place. But, I personally found it useful while working on another tab. So, a +1 from me for the proposal.
Also, in the question, linked, the users never told us what they found annoying. Was it that notification in the site or the indication on the tab, that was annoying? (In this regards, I think that the indication on the tab is atleast something that we require. I can refresh the browser if I felt like it.)