  In this paper, we study large-time asymptotics for the complex modified
Korteveg-de Vries equation \begin{equation} u_t + \frac{1}{2}u_{xxx}+3|u|^2
u_x=0, \end{equation} with the step-like initial data \begin{equation}
u(x,0)=u_0(x)= \begin{cases} 0, & {x \ge 0,}\\ A e^{iBx}, &{x < 0.} \end{cases}
\end{equation} It is shown that the step-like initial problem can be described
by a matrix Riemann-Hilbert problem. We apply the steepest descent method to
obtain different large-time asymptotics in the the Zakharov-Manakov region, a
plane wave region and a slow decay region.
