I know I’m late to the party here but I stumbled on this question while trying to diagnose my own alternator issue this morning.
When you see 13.8 on your battery with the car running that means the battery is fully charged. In solar power systems with 12v batteries 13.8 is called the float charge. Battery is fully charged and the charge controller is maintaining it at the prime voltage for battery health. When you see it up at 14.4 v it usually means that some load was on the battery between charges and the charge controller goes into boost mode allowing more power in to bring it back to full.