  In this paper, we are concerned with a a semi-discrete complex short pulse
(CSP) equation of both focusing and defocusing types, which can be viewed as an
analogue to the Ablowitz-Ladik (AL) lattice in the ultra-short pulse regime. By
using a generalized Darboux transformation method, various solutions to this
newly integrable semi-discrete equation are studied with both zero and nonzero
boundary conditions. To be specific, for the focusing CSP equation, the
multi-bright solution (zero boundary condition), multi-breather and high-order
rogue wave solutions (nonzero boudanry conditions) are derived, while for the
defocusing CSP equation with nonzero boundary condition, the multi-dark soliton
solution is constructed. We further show that, in the continuous limit, all the
solutions obtained converge to the ones for its original CSP equation (see
Physica D, 327 13-29 and Phys. Rev. E 93 052227)
