The experience you had with the dying man is exactly the same as I had with a dog we were operating on, I was monitoring the anaesthetic and during the operation the dog made various attempts to die, doing my job as best I could, along with the vet we managed to bring it through the anaesthetic and it survived. After returning the dog to the recovery kennels I was drawn back into the room and felt the dog dying and it wanted me by it's side, I sat holding it in my arms as it's life force left it's body and I felt very privileged that I could help it to pass.