[Bassasasin]

Could be a battery or connection... The old one may still have been bad. Alternators have cycles that are added together with diodes and when one phase goes out the voltage is a bit reduced.
Im thinking 13.8 is a good alternator.
A high current at just starting is somewhat OK because the battery has been discharged with the start process.

Batteries however have had their own wierd problems. Such as intermittent connections internally.

Alternator, battery, starter I consider are a combined set.. If you have a weak battery you can ruin your starter. If you have a bad alternator you dont charge your battery well and make it weak and ruin your starter.

Try cleaning up the terminals till shiney. Or have it done or at least checked.
If you have a battery charger put it on and charge up the battery. If you know what your looking for maybe during that process you may see a drain on the charger that seems to go on too long or too short and indicating the battery is in need of replacement.

Good Luck