What it sounds like to me. When the compressor stops there is a lot of torque on the housing. There are 3-4 rubber bushings that absorb the vibration of the motor running. One or more of these are wearing and need replacing. Or it might be something else, but I don't think so.
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if it's not the mount, its the compressor itself. With seals the motor itself, and seals wearing out - the system starts to finish its cycle with a bad amount of pressure in the wrong place, then when the power cuts, thunk - the pressure forces the motor to do something it doesnt like.
I'd go with redesign's idea - anything else, and you'll be in new compressor land.