Where does complex mathematics intervene in our real world? 
Answer: Twistor Theory! 
Twistors were introduced by Penrose [11, 13] in order to provide an alternative description of Minkowski-space which emphasizes the light rays rather than the points of space-time. Minkowski-space constructions must be replaced by corresponding constructions in twistor-space. The twistor programme  has met with much success:
The description of massless free fields (the Penrose transform)
The description of self-dual Einstein manifolds
The description of self-dual Yang-Mills fields
The description of elementary particles (rather tentative).