Try some smaller fixes before changing the alt, but they will all need them eventually.

Maybe your sensor that reads voltage is bad? Sounds like the alt is charging the battery.

Have you considered taking it to a shop for a diagnostic?

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2002 X5 4.6is - Titanium Silver / Black Alcantara - Dinan Stage 2 - Bilstein/Eibach - Brembo BBK ----SOLD---
1993 318i - Brilliantrot / Tan - bone stock daily driver