19c. British idiom for "die, be killed" (popularized during World War I), "probably from thieves' slang, wherein to go west meant to go to Tyburn, hence to be hanged, though the phrase has indubitably been influenced by the setting of the sun in the west" [Partridge]. Compare go south.
"vanish, abscond," 1920s, American English, probably from mid-19c. notion of disappearing south to Mexico or Texas to escape pursuit or responsibility, reinforced by Native American belief (attested in colonial writing mid-18c.) that the soul journeys south after death.