  We discuss the possibility that dark matter is made of a new complex massive
vector field with a global $U(1)$-symmetry. The field interacts with the
Standard Model via coupling to the gauge field of the hypercharge group. We
study classical homogeneous configurations arising in the theory. Some of these
configurations include magnetic components of the Standard Model
electromagnetic field tensor. This opens a possibility to relate the origin of
possible primordial magnetic fields to the cosmological evolution of vector
dark matter condensates composed of magnetic bosons.
