I’m not sure if it already supports this, but i doubt that it does, since they were only invented after the death of BeOS.
Would be good if Haiku could support push notifications…
There you go… my two cents… 
Haiku does have notifications, and a preferences pane to go with the feature, but they aren’t totally ready for widespread use yet by third-party app[lication] developers, like mature platforms currently offer in their SDKs (i.e. iOS, Android, Win10).
As apgreimann stated, Haiku is provided with a notification system: when You boot Haiku ,as example, you will see a notification about the activity of your connection: you will also see this notification when you setup an email accont on Haiku or when you restart the media services: you will be alerted about new messages/events.
Personally I use a lot this system in my bash scripts (eg with cron or with Filer, to perform various automatic operations).
There are also some third-party applications (eg some Qt applications) which rely on the Haiku’s notification system.
Some example:
