1985 735i Stalls

I recently aquired a 1985 735i with a little over 60K. The car has started to stall while driving. The engine shuts off and the warning lights come on...but the car will start again...sometimes almost immediately making a very jumpy ride. It's almost like a manual transmission jump starting itself back. It does it at any speed.

The fuel pump has been humming, so I'm thinking that may be the problem? I've had fuel pumps hum for years without failing. Is this symptom like those of a fuel pump failing?