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.)