  In this paper, we study large time behavior of complex-valued solutions to
nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity
in two spatial dimensions. To find a possible asymptotic behavior, we consider
the final value problem. It turns out that one possible behavior is a linear
solution with a logarithmic phase correction as in the real-valued case.
However, the shape of the logarithmic correction term has one more parameter
which is also given by the final data. In the real case the parameter is
constant so one cannot see its effect. However, in the complex case it varies
in general. The one dimensional case is also discussed.
