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.