Let F(z) be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural 7epsi;-clusters of roots of F(z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is twofold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of F are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper  and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schonhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.