Explain "hypersleep" from Alien, Prometheus, etc.?

How it might work in real life (in the far distant future):

Scientists like Mark Roth have studied suspended animation by starting with cases of spontaneous suspended animation, like the Canadian toddler who survived 2 hours of bitter cold in a nappy, the Japanese man who survived 23 days in the cold with no food or water, and the 29-year-old skier found frozen and revived after 9 hours. Here they have found a link between low oxygen and low temperature that triggers the body to slow metabolism. Scientist are also studying the effects of hydrogen sulfide, which the body makes in low doses, but when breathed in in large doses can cause the body to shut down completely - an effect that can be reversed with the reintroduction of oxygen. The thought is that perhaps if your body was producing a little extra hydrogen sulfide than normal, it might slow down the metabolism enough before freezing for you to survive the cold. You sort of turn yourself off before you get too cold.

Roth has successfully induced a 6-hr hibernation in mice using hydrogen sulfide, with successful reanimation, but experiments with larger mammals have so far been unsuccessful. The goal at this point is to increase survival rates in surgery by slowing things down a bit.

How hibernation actually works in animals:

• It isn't sleep. There are drastic physiological changes involved.
• Brain waves of hibernating animals closely resemble their wakeful patterns, though they are somewhat suppressed. The brain serves as a control over mechanisms regulating body systems.
• When an animal wakes from hibernation, it exhibits signs of sleep deprivation and may need sleep to recover.
• Preparation is essential for successful hibernation - primarily insulation and food stores (eaten intermittently in waking periods or eaten ahead and stored as fat).
• Animals have a minimum temperature (set point) below which their endocrine system will not let the body fall. The larger the animal the higher the setpoint, because it would require too much energy to reheat a large body. Some say that bears do not actually hibernate - they are too big to allow their temperatures to drop very far.
• During hibernation, the heart rate can drop to 2.5% of normal, breathing drops 50-100% (some reptiles stop breathing altogether), consciousness is greatly diminished, and body fat is used to maintain these minimal body functions.
• Fecal matter is not produced because nothing is passing through the digestive tracts. The body does produce urea, but it is broken down into amino acids rather than excreted. Hydration comes from body fat.

Extrapolated thoughts:

• I think the engineer would probably produce enough internal activity to register on equipment (if we allow that he could have survived at all).

• Brain waves that resemble waking brain waves indicate to me that there would not be dreaming.

• The body instinctively curls up to preserve heat. Sleeping supine or in the nude seems counter to animal instinct.

• This would be an artificially induced hibernation, so presumably they could reduce temperature below what fat stores would normally allow for because re-acclimation would be done artificially as well. Still, none of these film characters has any body fat, so I am not sure what sustains them.
• Studies of bears show that they maintain muscle mass, muscles strength, and bone density (likely because of rich fat stores). I am not sure how this would work for humans over the long haul. Bears also move around during hibernation, shifting position about once a day.
• To survive without fat stores, humans would need either a food source (which it seems would trigger digestive functioning), or rather than being in suspended animation, they would have to be actually frozen. This involves a whole other realm of science (cryobiosis), the probable replacement of blood with saline before freezing, and a complicated reanimation process that includes the return of blood and an electric shock. The engineer would not have registered on equipment in this state.

From the book of Alien ISTR that hyper sleep is just a very deep sleep and you dream in the same way as you normally would only more vividly due to the deeper sleep and longer time spent in it.

However, you do not need food or water (or at least do not seem to). Hence one can be in hyper sleep for long periods (Ripley managed 57 years) without serious ill effects and without aging. It works, basically, as a machine induced coma without the need for nutrition which also prevents aging (see the Ripley example).

Given the difference of time between Prometheus and Alien your theory on hypersleep placing the occupant in suspended animation could be correct as we know in Aliens that Ripley spends 60 years in hypersleep. Also in Aliens the scanners which break into the Narcissus to uncover Ripley do not appear to pick up her vitals from the scavengers dialogue when they enter they seem to not be aware of any lifeforms as they dust off the life pod. So Weyland Yutani may have looked into the Engineers technology as considering the Prometheus mission the company would have likely sent more teams to find out what happened and Weyland Yutani employees would have been given a mandate to investigate alien tech. As for Prometheus the comment above from Alien ISTR seems to explain the dreaming element. I do think you are thinking along the right lines though but all I could say is the Prometheus technology is less advanced as Alien is supposed to be but maybe Neil Blomkamp may include some more insight in Alien 5.