If $M$ is a real-analytic Riemannian manifold, then there exists a canonical "adapted complex structure" on a neighborhood of the zero-section in the tangent bundle $T M$. These structures were introduced independently by Guillemin and Stenzel and by Lempert and Szoke. It is possible to understand the adapted complex structure in terms of the "imaginary time geodesic flow." I will describe a new family of complex structures in which one modifies the problem by adding a "magnetic field" on $M$, described by a closed 2-form. These new complex structures are described in terms of the Hamiltonian flow for a particle in a magnetic field, evaluated (in a suitably interpreted way) at an imaginary time. For a constant magnetic field on $\mathbb{R}^2$ or $S^2$, the magnetic complex structure can be computed explicitly and is defined on the entire tangent bundle.