  Previous observations of the Ap star HD 32633 indicated that its magnetic
field was unusually complex in nature and could not be characterised by a
simple dipolar structure. Here we derive magnetic field maps and chemical
abundance distributions for this star using full Stokes vector (Stokes $IQUV$)
high-resolution observations obtained with the ESPaDOnS and Narval
spectropolarimeters. Our maps, produced using the Invers10 magnetic Doppler
imaging (MDI) code, show that HD 32633 has a strong magnetic field which
features two large regions of opposite polarity but deviates significantly from
a pure dipole field. We use a spherical harmonic expansion to characterise the
magnetic field and find that the harmonic energy is predominately in the
$\ell=1$ and $\ell=2$ poloidal modes with a small toroidal component. At the
same time, we demonstrate that the observed Stokes parameter profiles of HD
32633 cannot be fully described by either a dipolar or dipolar plus quadrupolar
field geometry. We compare the magnetic field topology of HD 32633 with other
early-type stars for which MDI analyses have been performed, supporting a trend
of increasing field complexity with stellar mass. We then compare the magnetic
field topology of HD 32633 with derived chemical abundance maps for the
elements Mg, Si, Ti, Cr, Fe, Ni and Nd. We find that the iron-peak elements
show similar distributions, but we are unable to find a clear correlation
between the location of local chemical enhancements or depletions and the
magnetic field structure.
