Perhaps the most significant example of Chris losing his identity was his decision to change his name, at least in his own mind. He chose to do this not through legal means, but simply by introducing himself to others; it was a way of becoming an entirely different person than the well-to-do college graduate he had been while living in society.
Driving west out of Atlanta, he intended to invent an utterly new life for himself, one in which he would be free to wallow in unfiltered experience. To symbolize the complete severance from his previous life, he even adopted a new name... he was now Alexander Supertramp, master of his own destiny.
(Krakauer, Into the Wild, Amazon.com)
A person's name is one of their defining features, and changing it, even symbolically, is not a light decision. As he traveled, he adopted more and more of the "Supertramp" personality, until he found himself in mortal peril. It is telling that when he wrote his last S.O.S. he used his birth name, perhaps realizing that he might find himself dead and forgotten; despite his efforts to lose his original identity, Chris wanted to be known and respected by others, not to be forgotten as a simple hobo with no purpose.