I think it would be nice if, for our notification inbox, we could delete notifications after we've read them. Just for organizational purposes. (And if that gets added, perhaps the filter by date function could be removed for it.)
I approve of this suggestion because I would really like to be able to manage what I do at work separately from what I do at home. I'm happy to deal with SO, P.SX and other work related SX sites at work, but many other SX sites are just for personal interest and it wouldn't be appropriate to deal with them on work time.
I would love it if Inbox notifications were only 'marked as read' when I actually clicked on them. At the moment, if you click on any of the Inbox items, they all stop being highlighted (are marked as Read in UX terms).
That way I would be able to ignore non-work related notifications during the day, and have all my non work notifications still highlighted when I get home. Similarly, while at home I could ignore work related notifications, and have them all still highlighted when I get to work the next day.
workaround: You can use the RSS-feed of a question instead of using the star, and your favourite RSS-reader will certainly have this feature