It's when you're anaesthesized for surgery that your consciousness goes away.

No dreams, nothing. You don't exist during those hours. You cannot account for them later; they are deleted.

When I woke up after surgery I felt that 1.5 hours had passed. But in actuality it was something around 4 hours since I went under. It turned out that 1.5 hours before I woke up I had been brought out of anesthesia and after that I simply slept. I didn't have any dreams at all, but at least my internal clock had started working. A clock presumably doesn't need consciousness, but there's definitely a qualitative difference between anesthesia and sleep. And as I sometimes suddenly wake up with a solution to some problem, with no recallable dream preceding it, my personal subjective opinion is that there's some level of consciousness going on even in sleep. That anesthesia experience was so very different from anything I had experienced before, with no recall whatsoever of the time that had passed.

How do you estimate time elapsed without external inputs? Experiments where people cut themselves from external stimuli seem to show that on the contrary, we don’t good internal clocks on this regard.

I nearly always "know" how long I have been asleep, unless there's been excessive drinking involved (and that would be long ago). "Normal" drinking doesn't seem to affect this. And for shorter naps (less than two hours) it's pretty accurate.

People probably do this based on experience. You know from the wall clock time that you slept, say 7 hours. So on a daily basis, you know what 7 hours of sleep feels like and from that you can extrapolate: if you think you slept about half as much, maybe it was 3 hours. Kind of thing.

Or that we just don't remember the dream states. Have you ever woken up and the dream rapidly fades away?

When I had surgery, I had very lucid dreams. And they were actually quite arousing. Afterwards I heard that it was actually common.