I'm currently using SMS to send communications to some people and email to send notifications to others.
I like SMS because of it's speed and accessibility, but email is more flexible and less costly.
I'd love to have the option to implement some sort of in-app messaging with push notifications. It would mean I can freely send communications to the phone of a user; they will receive a notification and simply need to open the app to view and respond to it. It would need to have the option of two-way communication to be fully functional, but even one-way only would be a great start.
You could argue that emails essentially do this, but email notifications are often muted or lost in the flurry of emails, whereas an app notification would be much harder to miss.