complementary \kahm-pleh-MENT-teh-ri\ (adjective) - Serving to complete or make perfect; of two or more things, serving to compensate for each other's deficiencies.
"Dr. Pharr's complementary medicine was intended to compensate for some of the deficiencies of mainstream medicine, while 'Dr.' Simpson's complimentary medicine merely involved saying nice things about the patient's disease."
From the Latin complementum, "that which fills up or completes," from the verb complere "to fill up, fulfill or complete." Complere is, of course, the origin of our word "complete." Both complementum and complere acquired different meanings when they passed into Spanish and Italian, when the sense of "fulfilling" became dominant - in particular they were used to indicate a fulfillment of the requirements of courteous behavior. When they arrived in English via this circuitous route, they gave us "compliment" and "comply."