  Main sequence stars and compact objects like white dwarfs and neutron stars
are usually embedded in magnetic fields that strongly deviate from a pure
dipole located right at the stellar centre. An off-centred dipole can sometimes
better adjust existing data and offer a simple geometric picture to include
multipolar fields. However, such configurations are usually to restrictive,
limiting multipolar components to strength less than the underlying dipole. In
this paper, we consider the most general lowest order multipolar combination
given by a dipole and a quadrupole magnetic field association in vacuum.
Following the general formalism for multipolar field computations, we derive
the full electromagnetic field outside a rotating quadridipole. Exact
analytical expressions for the Poynting flux and the electromagnetic kick are
given. Such geometry is useful to study the magnetosphere of neutron stars for
which more and more compelling observations reveals hints for at least
quadridipolar fields. We also show that for sufficiently high quadrupole
components at the stellar surface, the electromagnetic kick imprinted to a
neutron star can reach thousands of km/s for a millisecond period at birth.
