When you enable airplane mode, do you see that you are still connected to WiFi (and/or Bluetooth)? If not, then you shouldn't be getting any messages or anything like that. You could still have some local alerts for some things, like let's say reminders and the like, which are essentially only based on time and not on receiving some notification, but alerts related to receiving some sort of a notification over a connection shouldn't be getting through if you have airplane mode enabled and don't have WiFi and/or Bluetooth enabled separately.
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Not really certainly if WiFi is not disabled by default with airplane mode -- as I recall it still is. What does happen is that airplane mode remembers the last state of connections that you used while in it, so if at some point you enabled airplane mode and then separately enabled WiFi and/or Bluetooth then the next time when you enable airplane mode it will remember the state you had it in and will use that state (meaning WiFi and/or Bluetooth can remain enabled if that's what you had enable last time you used airplane mode).