(How) Can I push notifications from my Mac notification center to my iPhone? I have a 2015 MBP running OS 10.11, and an iPhone 6s with iOS 9. Can I push notifications from the Notification Center on the MBP to the iPhone? If so, how?

And I mean any notification. I'm not talking about synchronizing individual apps.
You can install Growl in your Mac, and install Prowl in your iPhone. Then setup the Prowl to connect to your Growl instance with the API. Finally you need to forward all OS X notifications to the Growl.