While Im a fan of the .44 Mag cartridge, I own a Ruger Super Redhawk and Browning B92 lever-action carbine and have hunted both elk and deer with the cartridge, I have to say it has its bad points as well as its good ones.
On the plus side, the cartridges are relatively inexpensive for non-handloaders. This hopefully translates to more shooting. Driving a 240g Speer JSP at 1880fps, the bullet delivers 1500fpe out to about 50 yards and 1000fpe out to 135 yards. It has plenty of the right stuff within its range the bullet diameter and weight do a lot to make up for low velocity and pass-throughs are not uncommon. (People will disagree as to what its range is, but I put it at about the 1000fpe mark.)
On the flip side, recoil can be a bit stiff in carbines but a slip-on recoil pad will tame it right down. (My Browning has a steel buttplate ouch!) Its not the rifle I want in my hands when a 200-yard shot if offered, something I would take with a 170g .30-30, so it may make sense as a first deer cartridge in some areas and not in others.
In AZ it is not what I would choose as a first deer rifle unless money for ammo was a major concern. Relatively inexpensive surplus ammo is available for the .308 Win and I would give it strong consideration. The .308 has a much greater reach and is a cartridge you will never outgrow 1500fpe out past 370 yards and suitable for anything up to elk or moose, and recoil is no greater than my .44 Mag carbine. I picked up a used .308 last year and wish Id gotten one 30 years ago.
My 2 cents.