late 14c., "having no deficiency, wanting no part or element; perfect in kind or quality; finished, ended, concluded," from Old French complet "full," or directly from Latin completus, past participle of complere "to fill up, complete the number of (a legion, etc.)," transferred to "fulfill, finish (a task)," from com-, here probably as an intensive prefix (see com-), + plere "to fill" (from PIE root *pele- (1) "to fill").
late 14c., "make complete, bring to an end, supply what is lacking; fulfill, accomplish," from complete (adj.) and probably in part from Latin completus. Related: Completed; completing.
late 14c., "to bring to an end;" mid-15c., "to come to an end" (intransitive), from Old French finiss-, present participle stem of fenir "stop, finish, come to an end; die" (13c.), from Latin finire "to limit, set bounds; put an end to; come to an end," from finis "that which divides, a boundary, border," figuratively "a limit, an end, close, conclusion; an extremity, highest point; greatest degree," which is of unknown origin, perhaps related to figere "to fasten, fix" (see fix (v.)). Meaning "to kill, terminate the existence of" is from 1755.
1779, "that which finishes or gives completion," from finish (v.). Meaning "the end" is from 1790. Finish line attested from 1873.