Thanks for your comments. As others have suggested, I do not think moving the outside compressor unit is possible. I walked around my neighbors house and he has a deck the full width of his house in the back and the other side of his house is completely planted with trees and shrubs. My neighor is elderly and I believe he lives with his son or daughter in a nearby state most of the time although I sometimes see he looking at the front door but I have never met him since I am new to the neighborhood. I would love (and would pay) to move this unit. I wrote down the serial number from the unit, which is a Carrier and went to their web site. This unit was manufactured in mid 1997. So it is not that old. Another neighbor behind me which has been there for awhile said his old (original) outside unit was extremely noisy.
While on Carrier's website I did notice they had an accessory which was a compressor blanket. I was seriously considering getting a price for this installed and contacting either my neighbor or his son and asking whether I can have this done. So if you really think this will help I may persue this.
Frankly, I am not sure what I hear is actually the compressor itself or the fan motor. An HVAC tech probably could tell me which it is.
I did see the thick Celotex foam insulation board at the local Home Depot and wondered whether this would help. I am willing to try anything.