Yeah, watercooling does stay on after you turn off the car. You can hear the pump running for a few minutes after turning off the car.
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Hmm, I haven't heard any electric pump on mine running after shutting off. I heard a soft, barely audible high pitched buzz once, but that lasted only a few seconds, so I'm not sure that would be what you're talking about. It was probably something in the DI fuel system. Also, the factory Workshop Manual doesn't show any external water lines going to the turbo bearings, so any water jackets, if they're there, would be plumbed into the main cooling system, which is driven by the mechanical water pump. Are you quite certain about this electric pump?
On the other hand I tend to baby the car the last few minutes before I reach my destination, and also allow for some cool down time after I come to a stop before switching off. Maybe I've never shut it down while it was hot enough to trigger the pump, if there is one?
My old Audi 5000 Turbo Quattro did have a separate electric pump for it's turbo bearings. It had it's own set of coolant hoses running to and from the turbo, and was very noticeable when it was on, because it also switched on the main radiator cooling fan. This was additional protection since the turbo coolant was shared with the main cooling system, so any heat absorbed from the turbo by the coolant would be more quickly dumped by switching on the radiator cooling fan. Good intentioned passers-by would always tell me my car was still running because it was so audible.