This idea has been developed and deployed to Canvas
TL;DR: The Canvas Teacher app should support the same notifications that the current Canvas Mobile (Student) app offers.
The Canvas Teacher app doesn't offer any push notifications for users of the app (mostly teachers). It would helpful if teachers using this app could receive and interact with notifications offered by the web and other Canvas mobile apps. This could lessen, or eliminate the reliance on notifications through email or the Canvas Mobile (Student) app.