I Like Leupold fixed power scopes. I use the FX-II in 4x32 and FX-III in 6x40. Both are lite weight and have very serviceable glass for legal light hunting in the woods. One of mine is a 1894 45 Colt, it wears the 4x. Stuffed with H110 and a 300g bullet I would think it very similar to your 44 mag. The 4x is plenty for any range those calibers are useable at.
Some type of low profile red dot. Theyre small and excellent to smack things with out to the effective range of a 44mag rifle. I put the cheap Bushnell trs 25 red dot on an Marlin SBL 44 mag and its a fun little carbine.
I have a 20mm Nikon Monarch African 1/4x on mine. German #4 reticle. This has proven perfect for my needs over the past 10 years. I am a heavy cover, short range hunter for the most part.
I have a Leupold 2.5x mounted just forward of the hammer on my 30AS and it is great. But, my favorite is a Vortex Venom on my 44mag. It is just so small and compact. Its not overwhelming on the small frame guns like the pistol calibers and I can see it perfectly with my old eyes.
Like so many have already said get a Leupold 1-4x or 1.5-5x, mount it as low as possible and have fun. They give the widest field of view when turned down low and all the magnification that a pistol caliber carbine can use when turned up. A fixed 2.5 x or 3x works very well too.
Goat
"I know you believe that you understand what you think that I said... But I'm not sure you realize that what you heard is not what I meant." ( A quote of my Father)