Complex fuzzy logic (CFL) is an emerging topic of research in fuzzy logic. Due to the circle structured codomain of the complex membership function, algebraic structure (including MV-algebra) in traditional fuzzy logic cannot be easily transplanted to CFL. Quasi-MV algebras are almost identical to MV algebras, except $\alpha \oplus 0=\alpha$ does not always hold. In this paper, our goal is to derive some algebraic structures for CFL. We first construct a quasi-MV algebra in complex fuzzy logic by introducing negation and truncated sum in the unit disc of the complex plane $S$. Next we construct a $\sqrt{\mathrm{\neg }}$ quasi-MV algebra over $S$ by adding an operation of square root of the negation. Moreover, implication connective and some derived connectives on $S$ are introduced. Furthermore, we construct a quasi-Wajsberg algebra over $S$ in which implication is a primitive connective. These algebraic structures are suitable for CFL.