I'm assuming that the .44 Mag limit is power wise, not a bore diameter limit, right? You prefer an autoloader, so we'll start there.
Find a semi-auto that fits your hand properly, regardless of what it is chambered for. Shoot before you buy if at all possible to see if it fits as well under recoil as it does when you just pick it up. After you've decided on your platform, pick a semi-auto cartridge that rings your bell (for a beginner, I say 9 or .45, but if you dig the .40, go for it), and that will decide which model of gun you actually get. Now, before you buy this gun, see if there is a .22 conversion kit available for it. The dealer may not have one, but should know if they are available. If there is, get the gun and the .22 kit. Shoot it with the .22 kit first, until you're used to the feel of the gun, then switch back to the main round and fire a few through it. For hunting small game, you can use the .22 unit, and for self-defense, you can use the primary cartridge. Just an idea...
Ultimately, the gun has to fit YOUR hand and be something YOU can handle. Now, you say you're adding to your arsenal, so I'll assume you've got some long guns and are not a novice shooter. If you're a novice handgunner, you still want to put in a lot of time with smaller stuff, even if you've been shooting a .300 Win Mag for 20 years.