The brief
The 420i had been showing a coolant puddle around the water pump area, the temperature gauge had started climbing, and a low-coolant warning had come up on the dash, with the cabin heater feeling weaker than it should. He brought it in before it overheated outright. Those symptoms point at the water pump. On these BMWs the water pump is electric, not belt-driven, and it pushes coolant around the engine and through the radiator to carry the heat away. When it fails it can leak coolant from its seal, which is the puddle, and it can stop pumping properly, which is the rising temperature, the weak heater and the low-coolant warning as the level drops. An engine that loses its coolant flow runs hot fast, and a hot BMW engine is one you stop driving.
The diagnosis
A pressure check traced the leak to the water pump, and a check of the pump confirmed it was failing electrically as well as weeping. The hoses, the radiator and the expansion tank checked out clean. That's a pump replacement, and on these the thermostat is the matching wear item in the same circuit, so it goes in at the same time, fresh seals on both, rather than being back in here when the thermostat lets go a few months later.
The work
The cooling system was drained, the failed electric water pump and the thermostat removed, and new genuine BMW-spec parts fitted with fresh seals. The system was refilled with the correct coolant, the air bled out the proper way following the procedure for the electric pump, and held under pressure to confirm the seals were dry. A road test confirmed the gauge sat steady and the heater warmed up on time.
The outcome
Gauge steady through traffic and at speed, no coolant traces, the level holding, and the heater working again. The 420i went home with the cooling system back to spec. An electric water pump gives little warning before it quits, and the failure at the end is an overheat that can cost a head gasket, so changing the pump and the thermostat together put the whole circuit right in one visit.