When your computer is in standby, it basically moves all your open files/programs into the system memory, and sends a trickle charge to it so the memory doesnt lose the data. But everything else in your computer is pretty much dead, so your cpu cant be processing information from AIM, or the internet connection.
So to answer your question, there isnt anything you can do in order to keep AIM "live" during standby, let alone any program. Even if your using a program that doesnt require an internet connection, such as encoding a video or tranferring files, everything gets stoped/paused when the computer goes into standby mode.