[holowoodman]

Depends on the "kind" of anesthesia, on the medication used and on the specific reaction of the patient to that medication.

For example there is stuff like Ketamine, which in some cases can live up to its other use as a recreational drug and give the patient very colorful dreams. There are sedatives that just take away the capacity to form memories, but leave you awake and aware, just calmer. In cases like some knee surgeries, it is possible to leave the patient fully awake, just paralyze and numb the legs.