My daughter's boyfriend has a 2003 Galant ES, 2.4L. They broke down several hundred miles from home and they were towed to Pep Boys to get the car fixed. The car had a dead battery that morning. They went by an autoparts store and they charged it and checked it and said it was good. So they headed home and as they were driving, the spedometer went crazy, a brake dash light came one, the battery dash light came on, radio would go off and on by itself. Finally the car died. That is when they were towed to Pep Boys. They said it was the alternator. They replaced it. THen they said it needed a new battery too. But they had already closed the register and said they would have to wait till the morning, or leave them the credit card number and they would charge it in the morning. Well they went over to Walmart, bought a new battery, put it in. It was still acting up. So they bought new battery cable clamps for the battery. Put them on and started it. The brake and battery dash light went off after a few minutes. They headed home. Well about 40 minutes later the brake and battery dash light started flashing on then off. Finally they stayed on. Another hour or so it died again. I went and put the car on a trailer and brought it home. The battery was down to 11.6 volts. I charged it for 30 minutes or so. It was up to 12.55 volts but I am sure it wasn't at full charge yet, amp wise. I started it and clipped my volt meter to the battery output on the back of the alternator and the neg. cable to the ground strap. It was only putting out 3 volts on the alternator. Before I started it, it was reading 12.4 volts. It sounds like a dead alternator to me. Is there any thing else I should check?

Make sure none of the big fuses are blown and that the alternator is getting power to it.

its good that you have bought new battery and alternator but make sure that wiring should done properly so that fuses work properly.

I believe your problem is related to the PCM shutting down your regulator through the C connection of the 4 pin regulator plug at the Alternator.
I found this with several Mitsubishi products that the PCM believes the engine idle is too low and to conserve the idle and preventing the engine from stalling the PCM will shut the Alternator down.
The 4 pins and their use are as follows
C= Computer connection/ used by the PCM to shut down the Alternator for fuel economy
FR=Field Report/ this connection is used to tell the PCM what the Alternators is doing
IG=Ignition /connection used to excite the alternator
L=Light /used to indicate the Alternator status to the driver via the battery light in the dash

Now a quick fix for your problem is to cut the C wire at the Alternator to prevent the PCM from shutting down the Alternator.
The other is is to check the idle specification and wiring so as the PCM does not get an eronious reading. There maybe other factors but I would start with this.

Check the relays and all fuses.