BE20 VISUAL SEPARATION IN USE CLB THROUGH OCCUPIED ALT HAD NMAC WITH CPR X. EVASIVE ACTION TAKEN. SEE AND AVOID CONCEPT.

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ATC ADVISED ACR X OF AN ACFT Y AT OUR 10-11 O'CLOCK POS; AT AN ALT OF 12000 FT IN A N DIRECTION. WE WERE TRAVELING IN A SW DIRECTION AT 13000 FT. ATC ALSO WANTED US TO RPT THAT TFC WAS IN SIGHT. WE ADVISED ATC THAT WE WERE LOOKING FOR THE TFC AND THAT WE HAD NEGATIVE CONTACT. AT THIS TIME; ATC ADVISED Y AT 12000 FT OF OUR POS AND ALT AND DIRECTION OF FLT. ATC ALSO WANTED Y TO 'RPT THE FALCON IN SIGHT.' WITHIN A FEW SECONDS; HE DID. AT THAT MOMENT; ATC ISSUED CLRNC TO MAINTAIN VISUAL SEPARATION WITH THE FALCON AND CLB TO AN ALT THAT I DID NOT HEAR CLRLY. ALL I KNEW WAS THAT THIS OTHER ACFT WAS CLBING THROUGH OUR ALT. WHEN THE CLBING ACFT WAS 5 MI AWAY; OUR TCASII ISSUED A TFC ALERT. THE TCASII DISPLAY INDICATED THAT THE CLBING ACFT WAS AT OUR 10-11 O'CLOCK AND 500 FT BELOW US IN A CLB. MY FO SPOTTED THE CLBING ACFT A FEW MOMENTS LATER AND ADVISED HIS EXACT POS FOR ME. I MADE VISUAL CONTACT; AND REALIZED THAT THE OTHER ACFT APPEARED NOT TO BE MOVING IN MY WINDSHIELD; INDICATING IT WAS NEAR OR AT MY PRESENT ALT AND ON A COLLISION COURSE. SINCE THE OPPOSING ACFT HAD IT'S RATE OF CLB ESTABLISHED; I BELIEVE A CLB BY OUR ACFT WOULD HAVE RESULTED IN A COLLISION WHEN THE 2 FLT PATHS MERGED. THEREFORE; A DIVE TO THE R WAS INITIATED. WHILE TAKING EVASIVE ACTION; ATC ISSUED A TA AND CLRED US TO DSND TO 10000 FT. WE NEVER HEARD; OR CLRLY HEARD; THE ATC ADVISORY NOR THE CLRNC TO DSND. I LEVELED AT 11000 FT TURNED BACK TO OUR ON COURSE HDG AND ASKED THE FO WHAT ALT WE WERE SUPPOSE TO BE AT. HE CALLED ATC TO VERIFY THE ALT. I CALLED INDIANAPOLIS ARTCC WHEN I LANDED. THE FIRST QUESTION I ASKED WAS WHAT TYPE OF ACFT WAS THAT AT 12000 FT. ATC REPLIED THAT IT WAS A BE200 (KINGAIR). I WAS TOLD HE WAS IFR. NEXT; I ASKED IF HE WAS CLR TO CLB THROUGH 13000 FT WITH US IN SIGHT VISUALLY. THE CTLR SAID YES; THE BE200 HAD SAID THAT HE HAD US IN SIGHT AND CLRNC TO 'MAINTAIN VISUAL SEPARATION WITH THE FALCON WAS ISSUED.' ANOTHER QUESTION I ASKED WAS; AT WHAT ALT DID THE 2 ACFT MERGE ON YOUR RADAR SCOPE? THAT REPLY WAS SOMEWHAT SURPRISING. ATC SAID THE NEAREST ANTENNA SWEEP THEY HAD ON US WAS ONE THAT PUT THE DA900 AT 13000 FT; 1.3 MI HORIZLY; ON A CONVERGING COURSE WITH THE BE200. THE BE200 WAS AT 12800 FT STILL CLBING. WITH HINDSIGHT BEING 20/20; I BELIEVE SEVERAL FACTORS TO BE CONTRIBUTING FACTORS TO THIS INCIDENT. FIRST; I BELIEVE CLRING AN ACFT TO CLB WHILE IN VMC CONDITIONS ABOVE 10000 FT WHERE SPDS ARE NOT RESTR TO 250 KIAS; DOES NOT MAKE A GREAT DEAL OF SENSE AND IS AN ACCIDENT WAITING FOR A PLACE TO HAPPEN. WHY NOT ALLOW VISUAL SEPARATION TO BE MAINTAINED IN THE FLT LEVELS ABOVE FL180? SECONDLY; DID THE BE200 PLT ACTUALLY SEE THE FALCON AT ALL? FROM OUR VANTAGE POINT; IT DID NOT APPEAR TO US THAT THE BE200 PLT SAW US; OR MAYBE HE LOST US IN THE CLB. WHATEVER THE REASON; THE BE200 PLT DID NOT CHANGE COURSE OR ATTITUDE TO AVOID US. LASTLY; TCASII COULD HAVE BEEN SMARTER. IT COULD HAVE ISSUED A RESOLUTION ALERT MUCH EARLIER BASED ON THE SPDS TO THE 2 CONVERGING ACFT. I HAD ACTUALLY RECEIVED VISUAL CONTACT WITH THE OPPOSING ACFT WITHIN 2 MI HORIZLY PRIOR TO TCASII ISSUING AN RA CLB COMMAND. AT THIS TIME I HAD NO IDEA WHAT TYPE OF ACFT WE WERE DEALING WITH. SEEING THE OPPOSING ACFT AT OR ABOUT OUR ALT; ALREADY ESTABLISHED AT AN UNKNOWN RATE OF CLB; I DIDN'T BELIEVE WE WOULD OUT CLB IT. CONSEQUENTLY; A DIVE WAS INITIATED. IN MY MIND; THIS ACTION PREVENTED THE MID-AIR COLLISION.