WX TURB RESULTS IN #4 ENG COMPRESSOR STALL RESULTING IN ENG SHUTDOWN.

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WE WERE CLRED TO CLB TO FL390. PASSING FL385 WE PASSED THROUGH THE TOP OF A WX FRONT WITH RESULTING MODERATE TURB AND POSSIBLE ICING. THE #4 ENG COMPRESSOR STALLED WITH RESULTING MAX EXHAUST GAS TEMP. THE ENG WAS SHUT DOWN IMMEDIATELY TO COOL DOWN AND FOLLOWED BY INFLT RESTART. IN THE PROCESS; HOWEVER; WE HAD TO BEGIN DSNDING TO MAINTAIN AIRSPD. AN IMMEDIATE 90 DEG L TURN OFF OF THE AIRWAY WAS MADE AND WE REQUESTED FL330. WHILE AWAITING CLRNC TO FL330 WE DRIFTED DOWN AS LOW AS FL357 AND WERE CLRED 'TO TURN SBOUND AND ONCE HDG S TO DSND TO FL330.' WE MAINTAINED FL357 UNTIL HDG S AND THEN DSNDED TO FL330. PASSING DOWN THROUGH FL350 A SUCCESSFUL ENG RELIGHT WAS ACCOMPLISHED. WE CONTINUED DOWN TO FL330; AND THEN REJOINED THE AIRWAY (B-586); AND RETURNED BACK ON COURSE TO THE N AND ON TO OUR ORIGINAL DEST WITHOUT FURTHER INCIDENT. AN 'EMER' WAS NOT DECLARED. WE WERE CLR OF CLOUDS DURING OUR ENTIRE DSCNT AND VISUALLY CHKING FOR TFC BUT STILL IT WAS UNCOMFORTABLE DSNDING WHILE INITIALLY AWAITING A CLRNC TO DO SO. IN RETROSPECT; I BELIEVE IT WOULD PROBABLY HAVE BEEN BETTER TO HAVE 'DECLARED AN EMER' AND LET ATC TRY TO MOVE TFC WHICH WAS AT FL350.