A B767 CREW; DEPARTING LAX; RECEIVED A PAX RPT FROM THE CABIN CLAIMING AN NMAC WITH ANOTHER ACFT.

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APPROX 15 MINS INTO FLT; FLT ATTENDANT RPTED THAT A PAX; WHO SAID HE HELD A PVT PLT'S LICENSE; THOUGHT WE HAD AN NMAC WITH AN ACFT ABOVE US ON CLBOUT FROM LAX. CONDITIONS WERE OVCST AT TKOF AND OUR TCASII WAS PLACARDED INOP. NEITHER THE FO NOR I HAD ANY SIGHTING OF ANY ACFT WHICH WAS UNUSUALLY CLOSE; AND ATC DID NOT ISSUE ANY ABRUPT AVOIDANCE COMMANDS. ATC DID; HOWEVER; ISSUE A COUPLE OF INTERMEDIATE LEVELOFFS AND GAVE INCREMENTAL TURNS BACK TOWARD THE SHORELINE. MY APPRAISAL IS THAT THERE MAY HAVE BEEN TFC ABOVE US; WHICH NECESSITATED THE RESULTING LEVELOFFS AND TURNS; BUT THAT THIS INDIVIDUAL'S PERCEPTIONS OF DISTANCE AND ACTIONS WERE NOT ACCURATE; AND MAY HAVE BEEN AMPLIFIED BY THE VISUAL SENSATION OF SPD AS WE CLBED OUT OF THE OVCST. THIS PERSON ALSO DID NOT IDENT HIMSELF OR TALK TO ME AT THE CONCLUSION OF THE FLT; SO I CONSIDER THIS TO BE A NON EVENT. SUPPLEMENTAL INFO FROM ACN 557444: WE ASKED THE FLT ATTENDANT TO INVITE THE PAX TO SPEAK WITH US AFTER THE FLT. WHEN WE DEPLANED; THE PAX DID NOT STOP TO SPEAK WITH US SO WE WERE UNABLE TO GAIN ANY ADDITIONAL INFO; INCLUDING HIS IDENT.