Wish this would have been in SM15, it's kinda the thing these days!  If I have my email client closed but Safari open to my Inbox it sure would be nice to get notification center notifications of new messages, i.e. toast notifications.  I think Chrome supports this now too.

I think Safari and others make pages go "inactive" or unloads them after X amount of time or when the phone is getting low on RAM. There is a good chance those notifications would only reliably get to you if you had the browser on the phone currently open. Without a dedicated app to receive push notifications it is unreliable.