Learjet 45 Captain reported a R ENG CMPTR FAULT amber CAS message illuminated in cruise. The flight crew performed an in flight shut down and returned to departure airport.

Narrative

I was acting as Pilot Flying on our second leg of the day from ZZZ1 to ZZZ2. We departed Runway XXR on runway heading up to 8000 ft. ATC turned us west and instructed us to climb to FL190. During climb through 15000 ft.; at an approximate location of ZZZ3 from ZZZ4 at 30 DME; the Pilot Not Flying and I heard a thump and felt a yaw to the right then stabilized. The R ENG CMPTR FAULT amber CAS message illuminated and amber indication in the N1 and the engine slowed to about 58% N1; all other engine indications remained normal. I continued to fly the airplane and took over radio communication while the pilot not flying performed the R ENG CMPTR FAULT checklist and continued climbing. After the checklist was complete we found reduced power on right engine; but all other parameters were normal. We decided the best thing was to return to ZZZ1. After we notified ATC we need to return we started to feel some vibration and reduced thrust to idle with little change in N1. Vibration increased; N1 slowed to about 30%. We elected to shut down that engine and request priority handling. Completed engine failure in flight and single engine landing checklist. Approach vectored us directly to a straight in visual for XXL. We performed visual to XXL with ILS guidance. Landing was uneventful. Priority vehicles were there and appeared to inspect our right engine then left.