Our 2001 Eldorado has a noticable sound when the ride control compressor runs, then it stops with a hissing sound that tapers off over a few seconds. Where should I look first?

 

My car does this also, which is normal. The suspension is filled with air. For example, if you have a car-load of people the motor will pump more air into the suspension to keep the car level. When everyone exits the vehicle the suspension will release the excess air causing the hissing sound. This is normal when you first start the car, you will hear the motor running then release some extra air. If you hear the motor running excessively for 7 minutes or more at a time, then you may have a leak which is not so serious, but should be taken care of. Hope this helped! =)