In the context of multiplayer gaming / MMORPGs, there's the term "multi-boxing", which seems to describe (albeit with a focus on a more specific niche) almost exactly what you're describing:
Multiboxing refers to playing as multiple separate characters concurrently in an MMORPG. This can either be achieved by using multiple separate machines to run the game or by running multiple separate instances of the game.
Multiboxing may be as simple as running two instances of the game on one computer (each logged into a separate account) and switching between them. Players may also use multiple computers, each with their own keyboard and mouse. This quickly becomes impractical with an increasing number of computers, so a keyboard demultiplexer may be used, which sends the signals from a single keyboard to several computers or virtual machines on one computer.
The Multiboxing FAQ (archived; original link is dead) further describes some specific variants on the terminology (like "2-boxing"/"dualboxing" or "5-boxing"/"quinboxing") and some of the technology used.
Another possibility, depending on the context, is to go for a pun, like "ambitechstrous". Not strictly proper English, sure, but if it catches on, it's only a matter of time before it ends up in dictionaries.