For a less expensive but very accurate alternative to the 460 S&W revolver option [which is not a bad suggestion at all BTW] you could purchase a 44mag Encore and have Mike Bellm rechamber it to 444 Marlin.
Mike's chambering philosophy and practice produces extremely accurate barrels, as Mike puts a true throat into the chamber which supports the bullet as it travels intop the bore, unlike the common practice of cutting a revolver type forcing cone in the throat area.
Here is a link to his website:
Bellm TCs Website
As well, Mike has a wonderful machined in brake that reduces recoil tremendously. His brake only takes up 1 1/8" of barrel length, has no bulge on it, does not interfere with the front sight, and cannot be misaligned with the bore since it is referenced off the existing bore. THe brake also employs an expansion chamber inside in addition to the gas venting holes you see in the sides, so gasses build up in the chamber against the slightly larger than caliber sized constriction at the opening, causing a much better braking effect than gas venting holes alone.
The 444 Marlin as a rechamber of an existing 44mag barrel is an appealing option when contemplating hunting with existing factory ammo.
In addition to factory loads from Remington and Hornady, CNC cartridge as well as other ammo firms like Buffalo Bore ammo produce really interesting loads in 444 Marlin as well.
Here is a link to the CNC Ammo website:
CNC Ammo
Here is a link to the Buffalo Bore Ammo Website:
Buffalo Bore Ammo
The length of the carteidge cleans up the factory chamber in the rechambering process, provides a match grade chamber, and you have the option of a machined in brake.
Here is a pic of the same machined in brake on a 44mag barrel Mike made me.