Putting someone in stasis goes far beyond a coma, though the issue of the mind is easily resolved actually. While it isn't known if or how people dream in a coma like state placing them in a medically induced deep coma is one way of making sure they don't endure years of some horrid dream world. In the case that mental activity is still a problem by the time we could perfect "stasis" I'd imagine we could develop drugs that could stop dreaming, after all a dream isn't some dark highly mysterious thing it's just a part of normal brain function.
Basic stasis already exists. During certain medical procedures, a heart transplant for example, the body is placed in a stasis like state and even for a time is technically dead, this allows doctors time to remove, replace and repair the heart and vessels without the body decaying and without the brain shutting down.
During this process the body's temperature is lowered, heart stopped and blood flow bypassed.
The problems associated with total stasis would probably be very similar to the problems encountered during cryogenics and the freezing of the dead to be revived at a later date. Of course with true stasis you wouldn't actually be dead.
Slowing bodily functions without causing harm (such as ice crystal formation which can be extremely harmful) is a very fine line. On the one hand you want to slow down the metabolic process but on the other you don't want to end up starving organs or shutting it down completely. It stands to reason that a large part of stasis is it's maintenance. The closer you are to death the more monitoring and work technology has to do.
Also since we really don't know if we have a "soul" should a person die (described as the cessation of breathing, heart function and the total degradation of brain function) it may be totally impossible to revive them. (I'm talking of course in the case of some problem and cryogenics) No one who has ever died 100% has ever been revived, clinical death is not the same as actual and total death.
Apart from the biological hurdles the question of technology would occupy a large part of research. How do we maintain a perfect system for years, free of human interference? The reason for stasis, usually, is to allow for interstellar travel without having a multi-generational crew. Which would mean that if anything went wrong it would be impossible to fix it from a "ground base" as the time for signals to come and go would reach years, and well death would occur way before that.
