Sudden loss of power
Are you sure the battery is being drained during this process? That would be a major draw from something, but then you said "...suddenly full power again,". It sounds more like a connection or a terminal working loose at times, not able to pass normal voltage. You sure need a voltmeter, my friend.
I think I would want to have that new alternator tested again, just to make sure it is not defective. A lot of aftermarket parts are turning up like that. Did you replace the alternator and battery because this condition was present? Do you know your battery cables and ground points are all good and secure, and corrosion free? You might want to double check all the grounds you can find- the battery negative cable, the engine to body strap, or cable, maybe an engine and computer harness ground point-look around the top of the engine for a ground wire coming out of the engine wire harness and going to ground at a nearby bolt. Also check all alternator connections.
If you have to keep looking, look at the power side of the power distribution scheme. From the battery, power goes to the Power Distribution Center, maybe called a junction box or junction block...it is where the fuse and relay block under the hood receives power, both from the battery and the tie-in from the alternator, and then distributes it to the major circuits for the car. With all your symptoms, it sounds like the problem is not far from the battery in the power distribution. Sayeth Dr. monte. Good luck.