I tried it, and even took a cape buffalo with a 9.3 JDJ in an SSK Handcannon. Didn't like it. (See the photo in the Barnes Reloading Manual #1. I was younger then.)
There are some gung ho handgun hunters that have had impressive success. J. D. Jones and Mark Hampton can shoot their SSK's as accurately at 200 yards as most rifle shooters. I was not one that ever gained enough confidence with my ability to place shots with a handgun. I sold mine and have stuck to rifles since.
Gary Phillips and Carl van Zijl of John X Safaris have guided many handgunners in RSA to some fine trophies. Boots is one famous hunter that comes to mind. Yep, that's his full name. SCI regulars probably know him.
The .44 mag will work, but you need to treat it like bow hunting.
Look for some old articles about handgun hunting from Bob Milek. He was the real deal. Not sure if he ever went to Africa, but I recall a G & A article years ago about a pronghorn hunt trying out some Nosler BT's. They seemed to work wonderfully for him, as I recall.