People often feel that a snake in a dream is something very negative, and yet this symbol is – just like the symbols of birth and death – one of the greatest symbols of transformation. Therefore, if a snake appears in your dream, you might want to investigate what is going on in your life on the inner and outer levels that might signify a process of transformation.
The symbolism of the snake – a creature that lives on the ground, in pits, under rocks, in caves, in the depths of the water – is that of the unconscious mind, and as symbolized by the snake it is something that is coming to life in the consciousness of the dreamer.
Joseph Campbell talked about the snake shedding its skin, referring to the newness that came underneath, and in that sense, a snake in a dream might signify a new beginning of some kind, after a loss of another.
For the serpent shedding its skin, to be, as it were, born again, is likened in the Orient to the reincarnating spirit that assumes and throws off bodies as a man puts on and puts off clothes.Joseph Campbell in Myths to Live By
The shedding of the skin was also considered a sign of immortality by ancient cultures. Campbell also uses the analogy to refer to the kind of rebirth that takes place with reincarnation, likening the shedding of the skin to the shedding of the current body for the next one in the next incarnation.
In Jungian dream interpretation, the snake is also referred to as the ouroboros which is an ancient symbol depicting a serpent or dragon swallowing its own tail and forming a circle. It has been used to represent many things over the ages, but it most generally symbolizes ideas of cyclicality, primordial unity, or the vicious circle. The ouroboros has been important in religious and mythological symbolism, but has also been frequently used in alchemical illustrations. (Source)
Finally, the serpent is also a symbol of healing. Not only in medical terms do we find the coiled serpent around the staff in the medical logo, but also the element of spiritual attainment as in spiritual yogic practice when the student’s kundalini rises up the spine, allowing him or her access to a greater spiritual understanding and growth.
Two excellent sources about snake symbolism in transformational dreams are: