We are all guessing here....Too many possibilities..

But you REALLY should put a multi-meter on it to see what volts are being generated while running.. YES, you should be getting a battery light if alt is bad, but that is too simple of an indicator and may not be working.

So, I would either get a multi-meter, a cig-lighter volt meter, or unlock test# 9 on cluster sub-menu to see how many volts are being produced while running, OR bring it in to a non-dealer BMW specialist for help.

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'03 530i - manual swapped - 425k miles
'06 325Cic auto 115k
'05 X5 3.0 auto 158k
'99 E39 528i 5-speed 130k