A B767-300 ON APCH TO HNL EXPERIENCED A HYD PROB AND TURNED OFF COURSE WITHOUT COORDINATING WITH ATC.

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WHILE ON A N HDG FOR A 90 DEG INTERCEPT TO THE FINAL APCH COURSE FOR RWY 8L IN HNL; WE GOT A L HYD QUANTITY MESSAGE ON EICAS. STATUS PAGE SHOWED ZERO HYD QUANTITY. WE TOLD APCH WE NEED A L TURN OUT OVER THE WATER AND I STARTED A L TURN. CTLR TOLD US TO TURN R TO INTERCEPT THE LOC. WE TOLD CTLR WE WERE TURNING L. CTLR TOLD US THERE WAS TERRAIN AND TO TURN L TO 180 DEGS IMMEDIATELY. WE ANSWERED THAT TERRAIN WAS IN SIGHT AND NOT A FACTOR. CTLR MODIFIED OUR HDG TO 230 DEGS AND SAID WE SHOULD HAVE TOLD HIM EARLIER. CTLR THEN IMMEDIATELY STARTED TALKING TO OTHER ACFT. BEFORE WE COULD GET A WORD IN TO DECLARE AN EMER; THE EICAS MESSAGE WENT AWAY AND THE STATUS PAGE SHOWED GOOD QUANTITY FOR THE L HYDS. WE TOLD CTLR WE WERE READY FOR THE APCH AND HE VECTORED US FOR APCH AND LNDG. I TURNED L OVER THE WATER TO TAKE TERRAIN OUT OF OUR SITUATION. ALSO; I WANTED TO STAY OUT OF THE HNL TFC AREA. GOOD PLAN; POOR EXECUTION. I SHOULD HAVE FOLLOWED CTLR'S INSTRUCTIONS; INTERCEPTED THE LOC AND THEN ASKED FOR VECTORS OVER THE WATER. INSTEAD OF REDUCING THE CTLR'S WORKLOAD; I ADDED TO IT. IT WAS THE WRONG DECISION AND IT WILL NOT HAPPEN AGAIN.