Will I dream?
No, the anesthetic state is more profound than the usual type of sleep in which dreams occur.
Coma is a state of profound, and usually, prolonged unconsciousness, meaning the patient cannot be aroused even by intense stimulation. The term, Coma, is usually reserved for unconsciousness caused by disease or injury, such as severe head injury or a metabolic disturbance such as very low blood glucose. I've got no idea if patients in a true coma can dream or not. This would be a question more fitting for a neurologist. The EEGs I've looked at (with a neurologist officially reading them) done on comatose patients are grossly abnormal and do not indicate the patient is in a state resembling natural sleep (they aren't in REM sleep, etc.).
Anesthesia is a loss of sensation, with or without unconsiousness (a common example of anesthesia without unconsciousness is a spinal anesthetic, where the patient has temporary loss of sensation below the level of the spinal anesthetic, but is completely awake and aware. Most people think of "general anesthesia," a reversible, induced state of unconsciousness along with loss of sensation, loss of awareness, an inability to move, and autonomic control (refers to control of heart rate, blood pressure, etc.). Dreaming definitely occurs for some patients under general anesthesia. Most patients I've talked with that tell me they had dreams report them to be pleasant. I always tell patients to think of something nice while they are drifting off to sleep so they will have good dreams and it seems to work—there is research suggesting you are able to influence patient behavior at this point (some studies show an improvement in smoking cessation in patients that listen to tapes with repetitive postitve statements about quitting smoking, etc.).
Awareness under general anesthesia is very rare—estimated from large studies to be 0.05-0.2% of healthy patients undergoing elective surgery. Most patients that experience recall do not report unpleasant experiences. Awareness is more common in emergency surgery such as that needed in severe trauma, or emergency Caesarean section under general anesthesia. This is when, in order to save a life, a decision is made to risk awareness to optimize the patient's chances. (You're too sick to tolerate any more medications, etc.) This is rare, and in this situation, a drug that blocks memory formation may be given as much as the patient can tolerate it. I don't know of any anesthesiologist routinely giving such a medication at the end of a case as such a drug is not known to produce what is called "retrograde amnesia," or erase previously formed memory. The benzodiazepines (valium, midazolam, lorazepam etc) certainly produce antegrade amnesia reliably and may be given as part of an anesthetic for many reasons, one of which might be to aid in preventing recall or awareness.
(BTW most patients "put under" for a tooth extraction are not receiving a general anesthetic, but are given sedation which would make it very likely that they would be aware, but not unpleasantly so, during the procedure.)
Critically ill patients in intensive care units frequently require sedation drugs for anxiolysis, and pain medications, and anti-psychotics or other such medications for agitation and delerium. As you can imagine, when you are seriously ill, your mind can play tricks on you and the most calm, mild mannered patient can exhibit atypical agitation, confusion, or delirium (a sudden reversible state of mental confusion). Many different drugs are used in an attempt to safely sedate patients so that they are comfortable, relaxed, and pain free. This is easier said than done. Some patients are in a coma because of their illness, others are deeply sedated and could be described as being in a coma due to the sedation required to provide care, but most patients are somewhere in between. This is because putting everyone into a deeply sedated state prolongs care unnecessarily and exposes the patients to additional risks associated with being ill in an ICU (hospital-acquired infection being the most common). It is recommended that every patient have their sedation stopped daily to allow them to awaken and be fully assessed neurologically. This is easier said than done but is a goal we are working towards.
I can't comment on your friend's experience due to patient confidentiality, but I can say that it is not uncommon for patients to recall negative experiences while critically ill. I would think being so sick is a negative experience in itself so it would unfortunately make sense that bad dreams or experiences occur. I would imagine most are misinterpretations of actual or perceived events, such as a bad dream you might have while ill with a fever, etc. We try our best to make the environment calm, quiet, gentle, and provide gentle compassionate care, and use drugs as needed to keep a patient safe, calm, and anxiety and pain free. This is an area of critical care that is incredibly challenging for a multitude of reasons.
The incidence of conscious awareness with explicit recall and severe pain has been estimated at less frequent than 1/3000 general anaesthetics [<0.03%]. Conscious awareness with explicit recall but no complaints of pain has been reported in the literature with an incidence of 05-2%.