A healthy charging system puts out up to 14.4V at idle w/no accessories on. Your voltage regulator seems to be trying to do its job since it seems to be pushing a bit more juice to the system when to start switching on accessories. Still, I think 13.8V or so is a bit anemic with the engine running and no real load on the electrical system.

Despite the previous rants, I do agree that the alternator (or components within) are suspect. I have later MBs and the only cure is swapping out the entire component, but yours is early enough to have the seperate regulator and brushes that can be replaced w/o tossing out the whole unit...far cheaper of course.

And a sick charging system can bring an already-old battery to its ultimate demise...

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2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car