I am a fairly good putter...and I sometimes practice putting not looking at the ball but rather at the hole. It is amazing how the eyes can communicate to the hands the direction and force required. There also is no head movement when using this putting drill. I have had friends even do this for an entire round of golf.
When player's eyes begin to anticipate the hit problems can arise. Most always it is because their practice of mechanics is lacking, however, the eyes need to be calm when you putt. If looking at the hole remedies this for a while....great!