yes, everything about this. for websites that still use forums (yes they are out there!) push notifications for quoted replies and direct messages would be extremely helpful. i believe push notifications are one of the largest reasons why forums died out in days of early smartphones - not having that instant feedback of interaction caused their own interactions to diminish as well.

the need for dedicated apps on your smart phone is now almost unnecessary at this point, unless you're doing something extremely niche. what a win.