Lagrange interpolation is extended to the complex plane in this paper. It turns out to be composed of two parts: polynomial and rational interpolations of an analytical function. Based on Lagrange interpolation in the complex plane, a complex variable boundary collocation approach is constructed. The method is truly meshless and singularity free. It features high accuracy obtained by use of a small number of nodes as well as dimensionality advantage, that is, a two-dimensional problem is educed to a one-dimensional one. The method is applied to twodimensional problems in the theory of plane elasticity. Numerical examples are in very good agreement with analytical ones. The method is easy to be implemented and capable to be able to give the stress states at any point within the solution domain.