On: Formal – Interruption free communication

[anExcitedBeast]

In Android, you can disable App notifications. That may accomplish what you're looking for.

[gcr → anExcitedBeast]

You can do that on iOS also. It's pretty fine-grained. The settings range from "no activity" (useful for Facebook) to "icon badge only" to notifications to "show a modal notification" (useful for my-flight-is-leaving-now airline apps)

[SomewhatLikely → anExcitedBeast]

There's also Do Not Disturb mode which turns all notifications, but also all phone calls, off for an adjustable amount of time.