Yes, Alabama like Arizona can require a driver to carry a SR-22 for various reasons. So if your previous state of Arizona mandated that you hold and continue to carry a SR-22 for a certain amount of time and that time period is not up then you will need to continue your SR-22 coverage in AL when you change over your driver's license to an AL license.
SR-22s are state specific though so what is required in one state may not apply in another state. When you move to Alabama, you will need to keep the Arizona mandated limits.
If you currently carry an SR-22 in one state but move to another state, you must fulfill the SR-22 filing period for your former state, even though you no longer reside there. Also, your insurance policy for your new state must have liability limits which meet the minimums required by law in your former (SR-22) state.
Arizona's minimum insurance requirements are 15/30/10 meaning $15,000 bodily injury liability for one person and $30,000 for two or more persons and $10,000 property damage liability.
In Alabama, the minimum limits of liability are 20/40/10 meaning $20,000 bodily injury liability per person, $40,000 each accident and $10,000 property damage liability.
Since Alabama's minimum coverages are more than Arizona's their minimum insurance plus the filing of the AZ SR-22 for you should take care of the requirements you have placed on your from Arizona. To be absolutely certain that you are doing what Arizona has mandated of you to keep your license valid you should check with the AZ DMV. They should be able to provide you with the information you need to give to your AL insurer so that your SR-22 is up to date until the end of your required filing period.