Desktop notifications would have (at least) 2 advantages:

1) It would give much quicker access to the critical alert information than a regular alert email, since you could layout/format the notification with perfect flexibility (as opposed to default email client notifications which devote most of the real estate to the "from" address, and subject and body in plain text)

2) With a couple buttons, the notification could give immediate access to actions such as "clear" or "mark as read"