I have just started noticing (not sure how long its been going on) that after turning off the ignition and getting out of the car, it sounds like the fuel pump in the back is still running for a while. Is this normal? Heck, is it the fuel pump? Before I get my knees dirty, what does anyone know.

That's a pressurized fuel system, could be releasing built-up pressure after you shut it down.

No ...
The Fuel pump will shut off.
The system will hold pressure....
It may be the fuel pump relay sticking.
It's located on the passenger side front. There is a removable panel at that location. The two relays side by side are the ones in question. One is the main power relay the other is the fuel pump relay . They actually work together so replace both. Very pricey….You might try to open them up and clean the contacts.
When you turn the key to the on position the pump will run for a second or two and pressurize the fuel system. When the engine starts it will come back on.

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2003 SLK 230 (85000)
1999 E300 Turbo Diesel (388,000)
1991 350SD Turbo Diesel (185,000)