While I don't consider the notifications to be part of the authentication - I agree, yes, notifications are good.
We're doing something like that at my company as well (its not a bank or anything). If you log from other countries, have too many repeated failed (or even successful) attempts in a time period, etc. you get notified on a dashboard/you can optin for an email (or /and an email that forwards to your SMSes).
then you can decide if/when you want to see it and to notify security if you think there's a problem.
saves a lot of time for the sec team as well which only will manually check when the alerts reach a higher threshold.