In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, 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 (Ling
et al
. 2016
Physica D
327
, 13–29 (
doi:10.1016/j.physd.2016.03.012
); Feng
et al
. 2016
Phys. Rev. E
93
, 052227 (
doi:10.1103/PhysRevE.93.052227
)).