We consider the asymptotic behavior of the complex-valued solution to the quadratic nonlinear Klein-Gordon equation in two dimensions. As in the real-valued case, we construct a solution which tends to a free solution with a logarithmic phase correction. However, the phase correction is determined by not only the nonlinearity but also the ratio of the amplitude of two waves reflecting the fact that asymptotic behavior of the solution to the free Klein-Gordon equation consists of two waves. This is a joint work with J. Segata (Tohoku University) and S. Masaki (Osaka University).