C525 INBOUND TO CRW EXPERIENCED TCASII EVENT WITH ANOTHER ARR ACFT.

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ON JAN/XA/03 AT APPROX XA00; I WAS ON AN IFR FLT FROM LOU TO CRW. I WAS HANDED OVER TO CHARLESTON APCH AND TOLD TO EXPECT VECTORS FOR A VISUAL APCH TO RWY 23 AT CHARLESTON. I WAS ASSIGNED A VECTOR OF 090 DEGS. I PASSED THROUGH AN OVCST LAYER AND WAS IN THE CLR AT 5000 FT. I WAS ABLE TO DETERMINE THAT THE ARPT WAS SOME DISTANCE TO MY R; AT 3 O'CLOCK POS; AND THAT I WAS MOST LIKELY GOING TO BE VECTORED ON A WIDE R BASE TO FINAL. I COULD NOT FIND THE ARPT VISUALLY AT THIS TIME. SUBSEQUENTLY; I WAS DIRECTED TO DSND TO 2600 FT AND TURN L TO 080 DEGS. I WAS VISUALLY SCANNING FOR TFC AND MY ONBOARD TA ISSUED A TFC ALARM. IT INDICATED TFC AT ABOUT 12 O'CLOCK POS. THE CTLR GAVE ME NO DIRECTION TO LOOK FOR THE TFC. AFTER I HAD PARKED AT THE FBO; I WAS CALLED BY SOMEONE IN THE RADAR ROOM AND ASKED ABOUT THE INCIDENT. I WAS ASKED WHY I WAS HDG 090 DEGS WHEN I WAS ASSIGNED A HDG OF 180 DEGS. I SAID THAT I WAS TURNING TO 180 DEGS AND THAT MY REPLY WAS THAT I WAS TURNING AND PASSING THROUGH A HDG OF 090 DEGS ON THE WAY TO 180 DEGS. HE SAID THAT A COMMUTER WAS GOING TO FILE A RPT BECAUSE HE HAD HAD TO DEVIATE ON THE BASIS OF A TCASII ADVISORY. HE ASKED ME TO CALL HIM BACK BEFORE I WAS DUE TO LEAVE. I DID CALL THE SAME INDIVIDUAL BACK WHO SAID THAT THEY HAD AN OPPORTUNITY TO LISTEN TO THE TAPE. IN ANSWER TO THE QUESTION 'HAVE YOU STARTED YOUR TURN?' I WAS HEARD ON THE TAPE ANSWERING 'AFFIRMATIVE; PASSING THROUGH 090 DEGS.' HE WENT ON TO SAY THAT THEY HAD TIMED THE PERIOD FROM THE INSTRUCTION TO TURN TO 180 DEGS TO THE POINT WHEN I REPLIED 'PASSING THROUGH 090 DEGS.' IF I HAD INSTANTANEOUSLY STARTED A TURN AND RESET THE HDG BUG FOR THE TURN; I WOULD ONLY HAVE BEEN ABLE TO TURN TO 110 DEGS IN THE PERIOD OF TIME. SINCE THE AUTOPLT MAKES TURNS AT STANDARD RATES; IT APPEARS THAT THERE COULD HAVE BEEN NO MORE THAN 2 OR 3 SECONDS DELAY FROM THE INSTRUCTION TO TURN TO THE RESETTING OF THE HDG BUG. NOTWITHSTANDING THE SPECULATION AS TO THE CIRCUMSTANCES; 1) I FOLLOWED INSTRUCTIONS FROM ATC. 2) I HEARD ONLY 2 PLANES ON FREQ AT THE TIME. I HAD NO ALTERNATIVE BUT TO FOLLOW ATC INSTRUCTIONS BECAUSE I DID NOT HAVE VISUAL CONTACT WITH THE OTHER TFC AND HAD NO CLR IDEA WHERE IT WAS. THE HURRIED INSTRUCTIONS TO ME ONCE THE EVENT BEGAN MADE IT DIFFICULT TO FOCUS IN THE COCKPIT LONG ENOUGH TO RESET THE MFD TO TFC MODE TO GET A MORE ACCURATE LOCATION OF THE OTHER TFC. I CAN THINK OF NO OTHER ACTIONS ON MY PART WHICH WOULD HAVE ELIMINATED OR LESSENED THE SIT.