The snake has more attested dream readings than almost any other image, and they have very little in common, which is itself the most useful thing to know about it.
The Babylonian Talmud, in the dream material of tractate Berakhot, gives an interpretation that no modern dictionary would predict. One who sees a snake in a dream has his livelihood readily available to him. If the snake bites him, his livelihood is doubled. If he kills the snake, he loses his livelihood. The logic runs through wordplay and association rather than through anything resembling emotional symbolism, and the outcome is economic.
Artemidorus distinguishes between kinds. The great serpent, the drakon, he associates with kingship, with time, and with wealth, on the grounds of its age, its guarding of hoards, and its association with certain gods. Ordinary snakes he reads as illness or as an enemy, and their behaviour in the dream governs the reading, what the snake does to the dreamer matters more than the fact of its presence.
Neither of these is about transformation, and neither is about the self.
The reading of the snake as a symbol of transformation because it sheds its skin is a genuine and old association in iconography, it lies behind the healing serpents of Asclepius and a good deal else, but that is a fact about how snakes were depicted and thought about while awake, not a documented tradition of reading them in dreams.