  A conjecture of Manin predicts the distribution of rational points on Fano
varieties. We provide a framework for proofs of Manin's conjecture for del
Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some
of our tools are formulated over arbitrary number fields.
  As an application, we prove Manin's conjecture over imaginary quadratic
fields K for the quartic del Pezzo surface S of singularity type A_3 with five
lines given in P^4 by the equations vw - xy = vy + wy + xz = 0.
