  Our aim is to find a complex continued fraction algorithm finding all the
best Diophantine approximations to a complex number. Using the sequence of
minimal vectors in a two dimensional lattice over Gaussian integers, we obtain
an algorithm defined on a submanifold of the space of unimodular two
dimensional Gauss lattices. This submanifold is transverse to the diagonal
flow. Thanks to the correspondence between minimal vectors and best Diophantine
approximations, the algorithm finds all the best approximations to a complex
number. A byproduct of the algorithm is the best constant for the complex
version of Dirichlet Theorem about approximations of complex numbers by
quotients of Gaussian integers.
