  Nonlinear Schr\"odinger equation, short pulse equation and complex short
pulse equation have important application in nonlinear optics. They can be
derived from the Maxwell equation. In this paper, we investigate a coupled
focusing-defocusing complex short pulse equation.
  The bright-bright, bright-dark and dark-dark soliton solutions of the coupled
focusing-defocusing complex short pulse equation are given.
  Then the breathers are derived from the dark soliton solution. The rogue wave
solutions are also constructed.
  The dynamics and the asymptotic behavior of the soliton solutions are
analyzed, which reveals that there exist the elastic or inelastic collision in
bright-bright soliton solution. But the interactions of bright-dark and
dark-dark soliton solutions are both elastic.
