  Interferometric radio astronomy data require the effects of limited coverage
in the Fourier plane to be accounted for via a deconvolution process. For the
last 40 years this process, known as `cleaning', has been performed almost
exclusively on all Stokes parameters individually as if they were independent
scalar images. However, here we demonstrate for the case of the linear
polarisation $\mathcal{P}$, this approach fails to properly account for the
complex vector nature resulting in a process which is dependant on the axis
under which the deconvolution is performed. We present here an improved method,
`Generalised Complex CLEAN', which properly accounts for the complex vector
nature of polarised emission and is invariant under rotations of the
deconvolution axis. We use two Australia Telescope Compact Array datasets to
test standard and complex CLEAN versions of the H\"{o}gbom and SDI CLEAN
algorithms. We show that in general the Complex CLEAN version of each algorithm
produces more accurate clean components with fewer spurious detections and
lower computation cost due to reduced iterations than the current methods. In
particular we find that the Complex SDI CLEAN produces the best results for
diffuse polarised sources as compared with standard CLEAN algorithms and other
Complex CLEAN algorithms. Given the move to widefield, high resolution
polarimetric imaging with future telescopes such as the Square Kilometre Array,
we suggest that Generalised Complex CLEAN should be adopted as the
deconvolution method for all future polarimetric surveys and in particular that
the complex version of a SDI CLEAN should be used.
