FLC OF AN MLG ACR ACFT RESPONDED TO TCASII RA AND THE VISUAL SIGHTING OF THE CONFLICTING ACFT RATHER THAN CLB AS INSTRUCTED BY ATC FOR EVASIVE ACTION.

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WE WERE BEING RADAR VECTORED BY DEN APCH CTL AND GIVEN A CLRNC TO DSND TO AND MAINTAIN 11000 FT MSL. APCH WAS VERY BUSY; AS USUAL; IN THE TERMINAL AREA. THE FO WAS FLYING WHILE I WORKED THE RADIOS. I MENTIONED TO APCH THAT I NOTICED A WDB CLBING OUT ON DEP AT OUR 1-2 O'CLOCK POS. THE TCASII SHOWED THE ACFT CLBING TOWARD OUR ALT. IT THEN STARTED A CLBING R TURN TO OUR DIRECTION. APCH CAME BACK AND SAID 'CLB IMMEDIATELY TO 15000 FT.' THE TCASII SHOWED RED IN THE CLB AREA OF THE VVI AND SAID 'MONITOR VERT SPD.' HAD WE CLBED EXPEDITIOUSLY TO 15000 FT MSL; WE WOULD HAVE HAD A CONFLICT WITH THE CLBING TFC. THE CTLR ISSUED WHAT WAS CONSIDERED THE CORRECT ACTION IN HER MIND; HOWEVER; IF WE HAD FOLLOWED THE CTLR'S DIRECTION AND IGNORED THE TCASII; WE WOULD HAVE HAD A VERY CLOSE CALL. WE MAINTAINED ALT ASSIGNED 11000 FT MSL AND TOLD THE CTLR WE WOULD MAINTAIN VISUAL SEPARATION WITH THE CLBING ACFT. THIS IS YET ANOTHER EXAMPLE AND PROOF PLTS SHOULD FOLLOW THE GUIDANCE OF TCASII WHEN FACED WITH A POSSIBLE CONFLICT; ESPECIALLY WHEN IN IMC CONDITIONS WHEN VISUAL CONTACT IS NOT ACQUIRED.