The many cases of Near Death Experience recorded would tend to show a conscious awareness even when the body is not functioning. Some people report hovering above their body or even going into other rooms and when they awake they know what happened in the other rooms etc.
There are IMO many levels of consciousness. Like when you are dreaming, you are only partially conscious. I read a study a while back that shows when the brain is completely unconscious it is completely unaware. It retains no memories of this period of unconsciousness. When the brain is not functioning there is no awareness. My experiences match this. Under anesthesia or the time I completely passed out, woke up on the floor with a bloody nose and sprained wrist. There was no awareness from the time I passed out to the time I woke up.
However, in these sub-levels of consciousness, a lot can be experienced in a dreamlike state.