In this talk, we will first introduce the complex continued fraction maps associated with some imaginary quadratic fields ($d=1, 2, 3, 7, 11$) and their dynamical properties. Baladi-Vallee analyzed (real) Euclidean algorithms and proved the central limit theorem for rational trajectories and a wide class of cost functions measuring algorithmic complexity. They used spectral properties of an appropriate bivariate transfer operator and a generating function for certain Dirichlet series whose coefficients are essentially the moment generating function of the cost on the set of rationals. We extend the work of Baladi-Vallee for complex continued fraction maps mentioned above. (This is joint work with Dohyeong Kim and Jungwon Lee.)