Under investigation in this paper is breathers and rogue waves on the plane wave/periodic background for the generalized complex short pulse equation. Firstly, the new Lax pair are obtained by means of a hodograph transformation, thereby enabling the construction of the generalized \((n,N-n)\) -fold Darboux transformation. As applications, a number of different types of breathers and rogue waves have been observed in plane wave/periodic background, including loop breather, eye-shaped breather, four-petaled breather, kink-breather, loop bright rogue wave, loop dark rogue wave, eye-shaped rogue wave and line rogue wave. Specifically, these results represent a novel contribution to the study of the complex short pulse equation, and we categorize these results, as well as to delineate the precise conditions under which they are generated. Subsequently, the dynamic behavior of these solutions is illustrated in great detail through the use of a multitude of images. We hope our investigation can provide a contribution that will facilitate the generation of additional novel localized waves on various backgrounds.
