According to The Information, Amazon is eyeing spoken push notifications for the Amazon Echo, and it could either be a really great or a really terrible thing depending on how the service is set up. For instance, if Alexa could offer notifications customized by time, place and type, it could perhaps let you know if the email you've been waiting for about that event coming up has arrived. It could alert you to terrible traffic conditions in relation to appointments you need to make.
That's looking on the bright side, however, as Alexa's spoken notifications could quickly bug users, even though it's an interesting suggestion for a feature. It would need a large set of customizational tools and permissions to work as flawlessly as it should in theory, but perhaps Amazon could figure it out.
Amazon hasn't yet offered any comment on the situation, but on a personal front it wouldn't be too obnoxious to have Alexa cutting in if her interjections are important. There's always an "off" button, after all.