We introduce a new approach to the local study of maximal surfaces in Lorentz–Minkowski space, based on a complex representation formula for this kind of surfaces. As an application we solve a certain Björling-type problem in Lorentz–Minkowski space and we obtain some results related to it. We also establish, springing from this complex representation, a way of introducing examples of maximal surfaces with interesting prescribed geometric properties. Further applications of the complex representation let us inspect some known results from a different perspective, and show how our approach can be used to classify certain families of maximal surfaces.