Complementarity is a concept in a number of fields:
In economics is a concept similar to that of externality. Two goods are complements if their cross elasticity of demand is negative. That is, as the price of one good increases, the demand to the other good decreases. When a large number of complementary products are interconnected, network externalities can arise. In general, complementarity exists when the decisions of other economic agents changes how I value things.
A good example of a complementarity is the example of keyboards. Virtually all keyboards use the QWERTY style, which was created so that typewriters wouldn't jam. Other styles include that promoted by Dvorak. Because QWERTY keyboards are so popular, an individual values it more than the one proposed by Dvorak, even though there is no reason to believe QWERTY to be inherently superior.
See also: network externality, cross elasticity of demand, list of economics topics
In Physics, "complementarity" is a basic principle of quantum theory, and refers to effects such as the wave-particle duality, in which different measurements made on a system reveal it to have either particle-like or wave-like properties. Niels Bohr is usually associated with this concept; in the orthodox form, it is stated that a quantum mechanical system consisting of a boson or fermion can either behave as a particle or as wave, but never simultaneously as both. A less orthodox interpretation is the "duality condition," described by the inequality due to Englert (see Phys. Rev. Lett., Vol. 77, 2154 (1996)), which allows wave and particle attributes to co-exist, but postulates that a stronger manifestation of the particle nature leads to a weaker manifestation of the wave nature and vice versa.
The emergence of complementarity in a system occurs when one considers the circumstances under which one attempts to measure its properties; as Bohr noted, the principle of complementarity "implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear." It is important to distinguish, as did Bohr in his original statements, the principle of complementarity from a statement of the uncertainty principle. For a technical discussion of contemporary issues surrounding complementarity in physics, see, e.g.,  (http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:quant-ph/0003073) (from which parts of this discussion were drawn.)
The Afshar experiment is claimed to question the validity of the principle of complementarity in quantum mechanics.