C wire Hooked to Compressor but terminated at Thermostat
Go ahead and use it

The fact that there is a C wire going out to the air conditioning compressor is normal where air conditioning is installed.  In fact, in systems with air conditioning where the C wire isn't clearly labeled, tracing the wires out to the compressor is a good way to find where to connect a C wire to.

So, go ahead and hook your blue C wire up to your new thermostat -- there is nothing else you need to do to have a functioning C wire.