This would be a nice feature. I'd request, though, that to whomever is going to work on notifications please don't bundle legitimate message alerts with "hey, use the app" notifications because it is the douchiest thing ever to do (looking at you Snapchat).
The lack of push notifications is a known issue that, as far as I understand, isn't solved so easily because it needs Lemmy to have some server side capabilities that it doesn't have (yet?).