I bind my mouse button 4 (which usually acts as the browser "back" button) to "Next Action". The Next Action key-bind is the same action as clicking on the notification button. And it was after using this for a while, that I realized, this actually lets me cycle between units.
On a new game, when my warrior and my settler is standing idle, I can just keep tapping on my mouse button 4 and the game will cycle between the two units back and forth, without ending their turns.
During normal play, you'll get other notifications mixed in the queue, like pick production, pick new tech to research, etc., but I think it works quite nicely.
NOTE, though, if you use Next Action on a unit that doesn't have full movement, it will end its turn. So now I'm always trying to keep my movement points tidy and use them up in one go.