A B737-300 AND TWIN CESSNA HAD NMAC WHEN DEPARTING ABQ.

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NMAC. WE TAXIED OUT FOR TKOF TO RWY 26 AT ABQ. THERE WERE TSTMS IN THE AREA WITH VIRGA EXTENDING S TO W OF THE ARPT. THE SURFACE WIND WAS 240 DEGS AT 15 KTS. TWR CLRED US ON HOLD. WE REQUESTED A HDG AFTER TKOF OF 300 DEGS TO AVOID THE WX; AND WERE CLRED FOR TKOF. TWR HAD A LIGHT TWIN ACFT ON FINAL FOR RWY 21; AND REQUESTED THEY FLY A FLT PATH BEHIND US TO LAND ON RWY 21. THEY SAID THEY WOULD COMPLY. WE DID NOT HAVE THE LIGHT TWIN IN SIGHT. TKOF; ROTATION AND GEAR RETRACTION WERE NORMAL. AS WE WERE CLBING THROUGH APPROX 250-300 FT AGL; I SAW THE TWIN AT 1 O'CLOCK POS; SLIGHTLY HIGH; DSNDING R TO L. I INITIATED A PULL TO THE STICK SHAKER TO PASS ABOVE. PRIOR TO TKOF; OUR PRIMARY ATTN WAS THE WX AND AVOIDING IT ON DEP; NOT ON OTHER ACFT IN THE TFC PATTERN. WE ASSUMED TWR AND THE LIGHT TWIN WOULD DECONFLICT OUR FLT PATHS. BUILDING SITUATIONAL AWARENESS ON OTHER TFC IS A BIG PLAYER AT ARPTS WITH XING RWY CONFIGNS. UPON ARR IN PHX; I CALLED AND TALKED TO ZAB SUPVR. SUPPLEMENTAL INFO FROM ACN 516796: ON JUN/FRI/01 AT APPROX XA51; OPERATING AS FLT XXX FROM ABQ TO PHX; WE WERE CLRED FOR TKOF ON RWY 26 AT ABQ WHERE WE ENCOUNTERED AN NMAC WITH ANOTHER ACFT LNDG ON RWY 21 AT APPROX 300 FT AGL. ACCORDING TO ABQ TWR; A TWIN CESSNA WAS SUPPOSED TO GO BEHIND US AND LAND ON RWY 21 BUT INSTEAD WENT STRAIGHT FOR RWY 21 AND CREATED A CONFLICT WITH US. I DID NOT SEE THE OTHER ACFT; BUT THE CAPT WHO WAS FLYING THE ACFT DID AND PERFORMED AN EVASIVE MANEUVER BY PULLING THE NOSE UP TO BTWN 20-25 DEGS AND HELD IT THERE FOR 10-15 DEGS; AT V2; UNTIL WE WERE CLR OF THE OTHER ACFT. THERE WERE TSTMS AND RAIN SHOWERS ALL AROUND THE ABQ ARPT; AND WE HAD BEEN DELAYED ABOUT 1 HR WAITING FOR FUEL BECAUSE OF LIGHTNING OVER THE ARPT. WHILE TAXIING OUT; WE HAD BEEN ISSUED A FLOW TIME WINDOW INTO PHX AND I BELIEVE THE TWR CTLR WAS PREOCCUPIED WITH US MAKING OUR FLOW TIME AND WITH ALL THE WX IN THE AREA THAT THEY WERE NOT MONITORING THE ACFT LNDG ON RWY 21. WE SPENT A FEW MOMENTS ON RWY 26 LOOKING AT THE WX ON THE DEP PATH ON THE RADAR TO DECIDE ON A HDG AFTER TKOF. THE TWR APPROVED OUR HDG REQUEST AND CLRED US FOR TKOF; AND AS FAR AS I CAN REMEMBER; DID NOT MENTION THE ACFT ON RWY 21. I DO NOT BELIEVE THAT WE WERE AT FAULT; BUT A COMBINATION OF THE OTHER ACFT NOT FOLLOWING INSTRUCTIONS AND THE TWR CTLR NOT MONITORING THEIR PROGRESS THAT WERE AT FAULT.