Suite.
Get the app ?

Install Suite on your phone and it opens full-screen from your Home Screen, with its own icon and a faster start. Installing is also what lets Suite reach you when it is closed: on iPhone, alerts are delivered to the installed app and nowhere else.

What reaches your phone

Suite interrupts you when something has actually happened and waiting for you to come back would cost you something.

You decide what gets through. Each kind above has its own switch in Account, under Notifications, and one switch at the top turns all of them off at once. Off applies to every device you are signed in on, not only the one in your hand, and nothing is lost: it all still waits for you inside Suite.