Pilot reported an engine failure on climb out. The pilot returned back to the airport and landed safely.

Narrative

Upon reaching 1;400 ft MSL the left engine started to slowly roll back. While checking in at 1;600 ft MSL the engine had fully failed. At the time the event occurred I had just finished turning off the fuel pumps. So; my first thought was that this was an engine driven fuel pump failure; however after not having much effect flipping to the high or low side on the pumps I suspected that may not be the case. I advised ATC and I calmly informed my passengers that we would be returning to ZZZZ. At this point I pulled out the QRH and began to run the engine failure after take off checklist. I should point out that before when the event occurred I did run through the memory items for an en gine failure after takeoff; except feathering it. I had a very light load of two passengers and somewhere around 420lbs of fuel or so. I made the split second decision to troubleshoot the problem. I had the altitude and the airspeed; as well as some time to get back to the airport. I didn't see it being a good choice to feather it until I had exhausted 'all' options. So; I went through everything fuel related; mixture; fuel selector; and fuel pumps. I don't remember exactly how many time I flipped back and forth from High to off; to Low; but it was many times. About the time I was on downwind for runway XX at ZZZZ; the engine came back online with the fuel pump on Low. After this a successful landing was made followed by a taxi back the gate. The engine continued to stumble and hesitate from idle anytime power was applied. Successful contact was made with crews which met us at the gate. The passengers were informed and I apologized to them for the incident; they were calm; and thanked me for getting them back on the ground safely. I am unable to determine exactly what caused this event; but it almost seems like some foreign debris may have been in the fuel lines. Something definitely disrupted the fuel flow; leading to the engine failure.