The voltage fluctuation is a classic example of a failing regulator or the lack of contact of the slip rings and the brushes, which in turn drives the regulator to fail.
Volt flux is a sure sign of alt failure that only gets worse. Tests are done at idle, should read over 13.5 volts and again at 2500 which should show an increase to >14.0v.

Quick test, with one eye on the voltage, bring the rpm to 2500 hold, tap the alt with a soft hammer and see what if any change.
At anything over 85k I would remove the alt, while checking for worn or frayed wires. If your plans are long term ownership buy new, if not buy reman AFTER making the seller test the unit before installing. Some remans are slow to break in and provide full output volts/Amps. As mentioned before some models have water cooled others are just air cooled. When you lay underneath your x you will be able to tell.
Good luck.

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2005 X5 4.8IS
The Blue ones are always FASTER....

Current Garage:
2005 X5 4.8is
2002 M5 TiSilver
2003 525iT
1998 528i
Former Garage Stable Highlights
2004 325XiT Sport
1973 De Tomaso Pantera, L Model
1970 Dodge Challenger T/A 4 sp Alpine White
1970 Dodge Challenger T/A 4 sp GoManGo Green
1971 Dart Sport, “Dart Light” package
1969 Road Runner 383
1968 Ply Barracuda 340S FB Sea-foam Green