You can add any normal email address in a somewhat roundabout way. You have to enable the email address for sign-in with Google
here. (If your other email is already a google email address, you don't need to do that.)
Once you have the email address enabled for sign-in, you can just add the email address as a user (in the users tab), and then sign in once as that user, go to the "me" tab, and enable email alerts.
(And, after doing that, you can go to the "me" tab using the account you used to activate, and disable the alerts for that email address.