Hi from Alaska-
There are 3 main hospitals in Anchorage- the Native Medical Center, which is run by a tribal consortium, Providence Alaska Medical Center, a Catholic not-for-profit, and Alaska Regional, which I believe is a Columbia/HCA facility. There are other hospitals across the state; many associated with the AK Native system, several associated with Providence, and I believe the Fairbanks hospital is run by a Lutheran not-for-profit? I'm not a nurse, but i work in long term care with LOTS of wonderful nurses.
Lots of people used to come to Alaska because the salaries were so good... but that's not really the case any more. The cost of living has really levelled out, and if you live in one of the major cities it isn't as expensive as living in LA or Boston or any one of many high-rent districts across the country. It IS expensive, however, to live in "the bush", and salaries generally reflect that.
People should come to Alaska because it's a beautiful place, a great place to raise a family, and filled with really nice, down-to-earth people. BUT- be prepared to freeze your ears off :) right now it's -15 here outside my little office.