  In this paper, we consider a blow-up solution for the complex-valued
semilinear wave equation with power nonlinearity in one space dimension. We
first characterize all the solutions of the associated stationary problem as a
two-parameter family. Then, we use a dynamical system formulation to show that
the solution in self-similar variables approaches some particular stationary
one in the energy norm, in the non-characteristic case. This gives the blow-up
profile for the original equation in the non-characteristic case. Our analysis
is not just a simple adaptation of the already handled real case. In
particular, there is one more neutral-direction in our problem, which we
control thanks to a modulation technique.
