I am currently not signed on many SE sites (but I am planning to). The thing is that the notifications are global from all the sites and it gets messy if you had a lot of activity in a short period of time (from different sites)
Wouldn't it be handy to have a feature to only check for notifications of the current site?. Needless to say this would be optional and turned-off by default.
Edit: I am aware that there is an activity tab, but it is hard to keep track of what you already read and is really not what I am looking for
Something like: