Alex Burgess is the son of Roderick Burgess, the man who had the temerity to plot to imprisoned Death, but captured Dream in her stead. Alex, though an indifferent student, learned magic from his father, and ultimately takes over his father's estate and magic order.
Afraid to free his father's prisoner, he kept Dream imprisoned, offering the entity the same deal his father had; trying to bargain immortality for release. Dream stone-walled the son as he did the father.
Alex became obsessed with Dream, to the point of reading the entry of Dream in the Liber Fulvarum Paginarum over and over again. This is the only magic book he read, and Alex eventually passes leadership of the Order on to Paul McGuire, who had been gardener at the Burgess estate.
Finally in 1988, Alex is a broken old man, and he threatens Dream to no avail. Alex's caretaker accidentally breaks the circle that holds Dream by pushing Alex's wheelchair over it.
Dream sees a chance at last, and collapses. When the guards come in to confirm his apparent death, Dream sends them to sleep and escapes to the Dreaming.
Weakened from years of imprisonment, Dream recharges himself. While Alex was taking a nap, Dream, in a cold rage, visits him. Alex pleads for his life, stating it was his father that imprisoned him, and that they wanted to imprison Death instead.
Dream reminded Alex that he didn't release him, and tried, just like his father, to bargain for things he shouldn't have and Dream couldn't give anyway. So as an act of vengeance, Dream gave Alex "eternal waking," in which he dreams that he keeps "waking up" from one nightmare to another.
Alex is stuck in the nightmare coma for years, until finally Dream finally releases him.