Yup, common side effect of a failing water pump, BUT as said, I would have the codes read first rather than just shelling out for a replacement.
Not a difficult job to DIY, and parts aren't too bad. I did one recently and a genuine pierburg pump plus a branded thermostat, BOTH with lifetime unlimited mileage warranty came to £370 inc vat. There are pierburg pumps on eBay for I believe around £240-250 but I thought £370 all in for both was a better deal, especially as if they ever fail again they will be replaced FOC.
Refilling the cooling system can be tricky, bmw insist on using a vacuum filling unit, which does the job correctly in minutes. I have only done them using the correct tool and automatic pump bleeding sequence, which runs the water pump for 12 mins continuously with the engine off and cold. But in order to do that you need a constant voltage supply connected to the battery as the ign, interior fan and dipped headlights need to be on for the full 12 mins. This draws about 40 amps, so you must have a diagnostic support unit attached to the battery that is capable of supplying the required current while maintaining 13v+
I can imagine it being a nightmare to vent the air from the system without the above.
Mike