Chris McCandless changed his name to Alex Supertramp because he wanted to leave every aspect of his former self behind.
When Chris McCandless became Alex, he was doing more than changing his name. He gave away his money, burned all of his possessions, asked his friend to return all his mail to sender, and left the modern world behind.
“Alex?” Gallien responded, fishing for a last name.
“Just Alex,” the young man replied, pointedly rejecting the bait. (ch 1, p. 4)
Names are one of our most powerful connections to the world. By changing his name to Alex (and later Alex Supertramp), Chris thumbed his nose at society and allowed himself to cut all ties, and begin his adventure on his own terms.
It is not until he was dying did he resume his name in the note he left for help. He “abandoned the cocky moniker he had been using for years, Alexander Supertramp, in favor of the name given to him at birth by his parents” (p. 198). At this point the adventure was over, and it was more important to give his family closure. In the note he reached out to civilization, at the end of his life.