The big game hunting in AZ is pretty good. If you archery hunt, there is over the counter opportunity for deer in the early fall and Dec/Jan time frame. Depending on where and what you want to put in for during the draws, there is plenty of opportunity for deer and elk. A good rule of thumb is once every 3-5 years for elk, and every year to every other year for deer (non trophy units/hunts). I manage to keep plenty busy from the early season from late August until Jan for big game.
There is duck hunting, but nothing like CA. It is very hit and miss, and you really have to work for it. It helps if you have a boat, but even then plan to travel and work hard for the ducks you get. Quail hunting can be good, and the dove hunting is usually great.