The short pulse equation is an important integrable equation in nonlinear optical fibers, which describes the propagation of ultra-short optical pulses in nonlinear media. Popowicz proposed the matrix short pulse equation and obtained four-component version of the short pulse-type equation in 2017. This paper focuses on constructing the binary Darboux transformation for the focusing–focusing, focusing–defocusing and defocusing–defocusing coupled modified complex short pulse (cm-CSP) equation, which can be reduced from the \(2\times 2\) matrix short pulse equation. By using a hodograph transformation, the cm-CSP equation converts to a coupled modified complex integrable dispersionless equation. Through the Darboux transformation, we derived bright–bright soliton, bright–dark soliton solutions, periodic-like solutions, rational solution and mixed solution (rational soliton) for the cm-CSP equation with vanishing and non-vanishing backgrounds. These solutions can be divided into smooth, cuspon and loop type. Some properties and asymptotic behavior of soliton solutions are analyzed. We must emphasize that compared with the traditional Darboux transformation, the binary Darboux transformation in this paper has the merits that the dark soliton and rational solution of the equation can be obtained more efficiently. Compared with the single-component modified CSP equation, the properties of the cm-CSP equation are more abundant, such as the fact that the cm-CSP equation has bright soliton and dark soliton solutions simultaneously. Compared with the focusing–focusing cm-CSP equation, the focusing–defocusing and defocusing–defocusing equation has some different properties, such as the existence of singular periodic-like solutions. We also emphasize that the asymptotic analysis of the collision solutions of the cm-CSP equation under the nonzero background has not been reported in other literature.
