Dal Longman Dictionary of Contemporary English, 4th edition (2005):
complexion n 1. [C] the natural colour or appearance of the skin on your face: Drinking water is good for the complexion. | pale/fair/ruddy etc complexion (=a pale, fair, red etc face) | fair-complexioned/smooth-complexioned etc She was fair-complexioned with blonde hair. 2. [singular] The general character or nature of something: Crime has risen under governments of every political complexion. 3. put a different/new/fresh complexion on sth to make a situation or event seem different: This document puts a different complexion on the matter.