Subaru - Outback :: 2004 - Motor Won't Stay Running When Car Started Intermittently?

I have a 2004 Subaru Outback. This fall I started having intermittent problems starting the car. It would run like a champ, no problems. Then every once in a while when I would try to start it, typically happens after it has sat all night, it would turn over just fine, motor would catch, then the motor would quit running (almost immediately upon start). It is kind of like the old days when your carburetor was not adjusted quite right and you would have to keep one foot on the brake the other on the gas while stopped at a stop light to keep the motor running.

The first time it happened we added "heat" as we thought it might be water in the gas line somewhere. It started, then ran just fine. This winter the problem started recurring more frequently. We added injector cleaner, it seemed to work a bit, but did not solve problem. We took it to the shop, and they could not figure out what was going on as nothing is registering on the computer and the car would start just fine for them. Brought the car home it worked OK for a week or so, then started up again.

The motor would catch then would quit running almost immediately. We were able to get it running good enough to get it to the shop (kept one foot on gas at stop lights, etc so would not kill). My car has been at the shop for two weeks now. It finally misbehaved on their watch, and they replaced a sensor, but they are not sure it fixed anything. They are keeping it so see if they can get a better diagnosis.

My first thought was the gas pump, but they said they had never seen a subaru gas pump fail and usually when a pump goes out, it goes out. They thought it could be electrical, but not sure as the problem is so intermittent. I would like to get my car fixed and back home. It's a great car; when it is running it runs great. The shop we work with is very good, and they are reluctant to start just replacing parts if it is not going to fix the problem.