About 3 weeks ago, I got my alternator replace. When I left the garage, they were no problems. Two days later, the battery light came on. Now, the battery light will randomly come on after I turn the car on. And it will randomly turn off after between 5 to 45 minutes of driving, depending on how fast I'm going. The codes I get are P0626 & P0620.

After turning the car on, it shakes and then the battery light comes on. If the battery light goes off, the car shakes again and all the electronics start working properly. Is it really the alternator, or could it be some cross wires, misconnected wires, bad terminals?

 

Yes check the wiring. Also check the ground wire to the block, and the belt too. Alternators don't go bad easily, it should last 20+ years. Voltage regulator, however, has a short lifespan as short as 5 years. If it's not replaceable standalone, that's when you replace the whole alternator.

 

I believe the folks who diagnosed the problem only did part of the job.
I also believe heiko offered a good suggestion, but I also believe you are going to need to find a shop with a good electrical tech.There is more going on here.

The PCM (vehicle computer ) controls the alternator and there is a good possibility the problem is caused by a loose or poor electrical connection somewhere in the vehicle.
Per your description of the car shaking is a tip off of possible loss of electrical power to the system. This is not something an ordinary shop will be able to locate unless the technician has a good electrical background and is very familiar with the vehicle electrical system.
That shaking is the engine trying to keep running but loss of electrical power is causing interruptions within the ignition system.
It could be the PCM,a poor connection or both.