  We give an account of the complex Arithmetic-Geometric Mean (AGM), as first
studied by Gauss, together with details of its relationship with the theory of
elliptic curves over $\C$, their period lattices and complex parametrisation.
As an application, we present efficient methods for computing bases for the
period lattices and elliptic logarithms of points, for arbitrary elliptic
curves defined over $\C$. Earlier authors have only treated the case of
elliptic curves defined over the real numbers; here, the multi-valued nature of
the complex AGM plays an important role. Our method, which we have implemented
in both \Magma\ and \Sage, is illustrated with several examples using elliptic
curves defined over number fields with real and complex embeddings.
