'Complement' is Latin for 'that which fills up' — from 'plere' (to fill), cousin of 'full.'
A thing that completes or brings to perfection; the number or quantity needed to make something complete; to add to something in a way that enhances or completes it.
From Latin 'complēmentum' (that which fills up or completes, a full complement), from the verb 'complēre' (to fill up, to complete, to fulfill), composed of 'com-' (together, intensive prefix) + 'plēre' (to fill). The PIE root is *pleh₁- (to fill), one of the most productive roots in Indo-European, yielding Greek 'plēthein' (to be full), Sanskrit 'pūrṇa' (full), Gothic 'fulls' (full), and Old English 'full.' Latin 'plēre' also produced 'plenus' (full), 'plus' (more), and 'plebs' (the common crowd
The words 'complement' and 'compliment' were once the same word. Both derive from Latin 'complēmentum.' The spelling diverged in the seventeenth century: 'complement' kept the original sense of 'completing,' while 'compliment' — influenced by Spanish and Italian 'complimento' (a courtesy, a fulfillment of social obligation) — shifted to mean 'an expression of praise.'