Well, that would largely depend on what size and type of wood you're going to be milling "constantly". If under 12" width and fairly soft wood, you could get away with a 70-cc class saw like a Husky 372 or Stihl 440/460, especially if it's got a modded muffler, though if you're spending a significant amount of time milling, for the extra money a 395XP or 660 would quickly pay for itself with faster cut times and less work on the part of the operator. If you're going to regularly see wood larger than 36" wide, or frequently work with harder woods, moving up to a 3120/880 (or an older 076 or 090) isn't beyond reason if production is a concern.
Personally I prefer the ergonomics and air filtration of my 395 for milling duty; however, the 066 has the chain tensioner on the side (front-mounted ones are blocked by the mill frame unless you modify it to allow access) and it also has a better exhaust location, so I end up using it more for my narrower pieces and for slabbing boards off of cants.
This was my freshly rebuilt 066 at work a few days ago cutting ~11" wide 1" boards from a Lodgepole Pine log:
http://www.youtube.com/watch?v=zZt4nFXKzsg
As you can see, it isn't the fastest process even with a fairly large saw. There's a pretty decent learning curve to it too - do a lot of reading here (especially the Sticky at the top of the forum) and learn from our mistakes! I milled for a full year and a half on my own before discovering this place. Looking back, I could have saved myself a ton of work and time knowing what I know now from talking and sharing ideas here. It's like free tech support but better!
And to answer your last question, yes, it's definitely possible and practical to cut beams and timbers with an Alaskan. In fact much easier than making boards, because fewer cuts are involved. It just might not be very
profitable, unless you have a custom market that will pay a premium. You can sure save a ton of money if you have time to spare though. Here are a couple threads of mine with some pictures of making boards and beams/posts out of the same log:
http://www.arboristsite.com/showthread.php?t=107756
http://www.arboristsite.com/showthread.php?t=76422
And a thread about the deck I built in 2008 out of Douglas Fir timbers that milled myself:
http://www.arboristsite.com/showthread.php?t=73147
I figure those beams would have run me well over $1000 at one of the smaller custom sawmills around town, and all they cost me were maybe $60 in fuel & oil for the saws and ATV. And a couple days' work cutting them and hauling them out of the bush, but that's half the fun anyway!