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I would assume alternator or starter. Starter seems to work normally - not sure if the solenoid can short without the starter trying to crank.
That leaves alternator. Anyone ever have a temperamental alternator?
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Also, the battery did go down to 12.3V (dash display) even though I was litterally cruising down the highway. I would have expected the voltage to be at least 13.6 even after cranking a few times.
When the car is running I would expect 13.7 - 14.7 volts. Especially with a new battery and not running A/C (guessing here).
When I was having issues, there was a post about checking voltages. I would expect higher than what you have.
Ever had the alternator tested? I would test and smack it with a plastic hammer or screwdriver butt while testing.
Another thought is the MAF or ECU connections. Unplug, inspect, plug.