He is the son of Roderick Burgess, mother unknown (but probably Ethel Cripps, and therefore half-brother of Doctor Destiny). He is taught by his father, and takes part in his rituals. On Roderick Burgess' death, Alex inherits his estate, including his magical order. He keeps Morpheus imprisoned, like his father trying to bargain for power and immortality in exchange for his releases. The order enjoys a resurgence in popularity in the 1960s, but by the 1970s it declines again. Alex passes the order to his boyfriend, Paul McGuire (formerly a gardener at the estate), and becomes obsessed with his prisoner and with his father. Finally, in 1988, Morpheus' guards fall asleep, and Morpheus escapes. He puts Alex into a nightmare of 'eternal waking'; he is forever dreaming he is waking up, and each waking degenerates into another horrible nightmare. This nightmare ends only with Morpheus' death in the ninth collection of issues in the series, The Kindly Ones.
Alex is quite tall and short-sighted. He has brown hair which he wears in a variety of styles throughout his life, but by old age he is bald and has come to resemble his father very closely. As a sidenote, his name almost certainly derives from Anthony Burgess's A Clockwork Orange (the protagonist of which is named Alex). His relationship with McGuire is deep and heartfelt, but his obsessions with his father and with Morpheus eventually come to rule his life. In The Wake, he appears again as the child that we see on his first appearance; since this is within The Dreaming, we may conclude that this is his true self, or at least how he sees himself. This may also reflect that the capture of Dream coincides with the time that his father, and his father's captive, begin to dominate Alex. Hence Alex's self-image stops growing at this age.
Alex is in many ways a tragic figure, perhaps the first statement of the theme that Desire explores in The Wake : "The bonds of family bind both ways". Had Alex not been born the son of his father, inheriting the imprisoned Dream, his life would probably have been much happier.