It is called the 357-44 Bains-Davis, and was developed for use in revolvers, typically 44 Mag revolvers with a 357 barrel installed. They avoided the issues caused by cases backing out and/or having to fit a new cylinder by fitting a plastic ring around the neck of the cartridge to totally fill the 44 Mag chamber, but there have been some made fron 357 revolvers with no issues (the straight-wall case does not back out like a tapered case would). It is a hot little number, one of my favorites. You can get almost 35 Remington performance out of a revolver.
Lotsa people make lotsa wildcats. I built a single-shot rifle for a guy about a year ago chambered for 22-44 (44 Mag case necked down to 22 cal). He had a reamer and dies made, and it shoots great. But there are all kinds of ways to get your jollies, just sit and think a bit.