The complex multiplication method is well-known for elliptic curves, where it may be used to construct curves used in primality proofs or to implement crytosystems, in particular pairing-based ones. A similar approach is possible for abelian surfaces, that are Jacobians of genus 2 curves, with considerable number theoretic complications. I describe an algorithm using complex floating point approximations with an asymptotically optimal running time, that is, quasi-linear in the size of the class polynomials produced as output. Our implementation has been used to carry out parallelised record computations and I present experimental data.