My Chrysler 300c keeps humming lightly when the car is off any ideas

Maybe the computer (ECM/PCM)? They stay on up to around 30 minutes after the car shuts off. Try to figure out what area it's coming from.

MechanNick is probably correct. So disconnect your battery and see if the humming stops. If it keeps humming, then it is not electrical. Sometimes turning a car off with A/C, you will hear refrigerant gurgling just a few minutes.

Question?? My 2005 300c is humming while car is turned on and while driving vehicle?? Why??

You can repost your question as an original post so more people will see it. I would guess it is your power steering fluid is low. Check transmission fluid too. These pumps whine when low on fluid. Probably nothing serious.