Buffalo Bore claims three feet of penetration through tissue and bone with their Heavy .44 Mag load, a 305 gr LBT LFN gas check bullet at 1325 fps from a handgun, 1700+ from a LA carbine. That should reach the brain/upper CNS of any animal, maybe excluding elephant. Someone who can shoot that load accurately and coolly and is well acquainted with the anatomy of the animal should be well served by that load. The second shot will not follow quickly for most of us, as recoil is stout.
A 525 gr hardcast load for the 500 Linebaugh is available from Buffalo Bore, at 1100 fps. Without special tacky gloves, this load is beyond my strength in the Ruger Bisley that John made for me, as I can't keep the revolver from rotating in my grip to strike the web of the thumb with the hammer, violently. It's painful to shoot and it rips flesh. This load provides a 21% increase in mass/cross section over the 305 gr 44 Magnum round, so should show somewhat better penetration. Bigger stronger men than me might not have any difficulty shooting this load. The more standard 440 gr load can be handled, and the 900 fps 440 gr "practice" loads will kill anything the Americas have to offer.
With a little practice, I think I can have good first shot accuracy with that 525 gr 500 linebaugh load. If I anticipated hunting in Africa and needed a sidearm, I'd consider the 500 Linebaugh over the 44 Mag, or, maybe better, the 475 Linebaugh. The 454 Casull packs a punch, but the bullet weights are less than the Linebaughs, greater than the 44 Mag.
A 4 or 5 inch barrel 500 S&W would be ideal, but its heavy to carry. I've handled but never shot one; it feels surprising good in the hand. Penetration with a given hardcast handgun bullet is proportional to velocity up to a threshhold velocity, above which more velocity does nothing or may give reduced penetration (due to bullet deformation). The threshhold is 1300 fps to 1400 fps, If I remember correctly. So its not immediately clear that the 500 S&W offers more for up close work than the 500 Linebaugh.
I think your 44 Mag would prove a comfort even in Africa. You're not likely to encounter a raging bull elephant in most places if you stay out of deep brush, unless you're looking for it. The 44 would handle lion to black mamba if you can shoot it quickly, might work on hippo or buffalo. Not the weapon of choice, no handgun ever is, but a whole lot better than a handful of nothing!