  Let $F(z)$ be an arbitrary complex polynomial. We introduce the local root
clustering problem, to compute a set of natural $\varepsilon$-clusters of roots
of $F(z)$ in some box region $B_0$ in the complex plane. This may be viewed as
an extension of the classical root isolation problem. Our contribution is
two-fold: 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
(Becker et al., 2018) and are based on the Pellet test, Graeffe and Newton
iterations, and are independent of Sch\"onhage's splitting circle method. Our
algorithm is relatively simple and promises to be efficient in practice.
