DC9 AND CANADAIR CHALLENGER HAD NMAC ON VISUAL CLB.

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ON CLBOUT WE WERE ASKED IF WE HAD THE CHALLENGER IN SIGHT. WE DID. (THEY WERE A GOOD DISTANCE AWAY AT 2 O'CLOCK POS; MOVING R TO L.) WE WERE TOLD TO MAINTAIN VISUAL SEPARATION. RAPCON THEN ASKED THE OTHER ACFT IF THEY HAD US IN SIGHT. THEY DID. THE CTLR TOLD THEM TO ALSO MAINTAIN VISUAL SEPARATION AS WE WOULD BE 'CLBING THROUGH' THEIR ALT. WE WERE THEN CLRED FROM 8000 FT TO 10000 FT. LEVEL AT 10000 FT FOR APPROX 20 SECONDS; WE BEGAN TO WONDER ABOUT ALT SEPARATION BASED ON WHAT WE SAW. WE WERE THEN TOLD TO CONTACT CTR. THEY DID NOT ANSWER ON FIRST CALL. ON THE 2ND ATTEMPT WE WERE GIVEN A CLB TO EITHER FL210 OR FL230. DURING THIS TIME IT BECAME APPARENT THAT WE WERE BOTH AT 10000 FT. THE CAPT MADE AN ABRUPT PULL UP AND I ESTIMATE THAT WE MISSED BY 500 FT OR LESS. ON INITIAL CALL OF TFC POS TO US THE CTLR MIGHT HAVE (PROBABLY?) TOLD US THEIR ALT; HOWEVER; WHEN HE TOLD THEM THAT WE WOULD CLB THROUGH THEIR ALT AND SUBSEQUENTLY CLRED US FROM 8000 FT TO 10000 FT; I ASSUMED THEY WERE AT 9000 FT. (WHY CLR US TO THE SAME ALT?) ALL THEY HAD ON WERE POS LIGHTS AND STROBES AND AT NIGHT IT'S DIFFICULT TO JUDGE ALT SEPARATION UNTIL CLOSER IN. (ADDITIONALLY WE WERE DISTRACTED TRYING TO RESET A MALFUNCTIONING DME.) THINGS THAT COULD HAVE HELPED: 1) MORE DILIGENCE BY BOTH CREWS AS TO ALT ASSIGNMENT. (WHY ASSIGNED SAME ALT IF CONVERGING COURSES?) 2) DON'T ASSIGN SAME ALT. (I DON'T KNOW IF CTLR MEANT TO OR IF IN ERROR. KIND OF A VICIOUS CIRCLE WITH #1.) 3) TCASII IN FREIGHTERS. WHILE THIS WAS NOT A VERY NEAR MISS; IT DEFINITELY WAS AN EYE OPENER.