There are two versions of the AIM app in the App Store right now. The free one, with ads, and the $2.99 one, with no ads. They both have push notifications.
The app does what was touted this year at the iPhone 3.0 event: display notifications when the app is closed, show you how many outstanding messages you have and basically keep you "connected" to the AIM service even when you're not actively using the app. The free version is here and the pay version is here.