Old English swimman "to move in or on the water, float" (class III strong verb; past tense swamm, past participle swummen), from Proto-Germanic *swemjanan (cf. Old Saxon and Old High German swimman, Old Norse svimma, Dutch zwemmen, German schwimmen), from PIE root *swem- "to be in motion."
The root is sometimes said to be restricted to Germanic, but possible cognates are Welsh chwyf "motion," Old Irish do-sennaim "I hunt," Lithuanian sundyti "to chase." For the usual Indo-European word, see natatorium. Sense of "reel or move unsteadily" first recorded 1670s; of the head or brain, from 1702. Figurative phrase sink or swim is attested from mid-15c., often with reference to ordeals of suspected witches.