ACR MLG TRACK DEV AFTER DEP FROM MDW.

Narrative

MDW TWR CLRED OUR FLT FOR IMMEDIATE TKOF FROM RWY 04 AND ISSUED DEP INSTRUCTIONS OF 'TURN L 260 DEG.' OUR DEST WAS DET AND I REALIZED THE DEP INSTRUCTIONS WERE PROBABLY INCORRECT FOR OUR DIRECTION OF FLT. BUT BECAUSE OF LNDG TFC TO RWY 04 BEING CLOSE IN; WE TOOK OFF AND DECIDED TO RECTIFY THE DEP AIRBORNE. WE SWITCHED TO CHICAGO DEP AND CHKED IN WITH 'L TO 260 DEG BUT WE'RE HEADED FOR DETROIT.' DEP SAID; 'THERE'S SOMETHING WRONG HERE; KEEP THE TURN COMING.' WHEN PASSING THROUGH APPROX 120 DEG DEP ASKED WHAT WE WERE DOING AND WHAT HDG WE WERE PASSING THROUGH? AFTER OUR REPLY HE SAID THAT WHEN HE SAID 'KEEP THE TURN COMING' WE HADN'T REACHED 260 DEG YET AND HE MEANT KEEP TURNING TO 260 DEG. WE WERE THEN GIVEN A R TURN BACK TO THE N. I COULD HAVE QUERIED TWR AS TO THE DEP INSTRUCTIONS BUT THERE WASN'T TIME PRIOR TO TKOF. POSSIBLY IMMEDIATELY AFTER TKOF BUT THE COCKPIT IS RATHER BUSY THEN. I COULD HAVE FURTHER CLARIFIED THE TURN WITH DEP BUT AT THE TIME THE INSTRUCTIONS SEEMED PRETTY CLR; ESPECIALLY SINCE A CONTINUED TURN WOULD PUT US ON COURSE. FROM OR STANDPOINT THERE WAS NOT A VISIBLE CONFLICT; BUT I HAVE NO IDEA WHAT THE CTLR HAD ON HIS SCOPE.