  As the remnant of Supernova (SN) 1987A has been getting brighter over time,
new observations at high frequencies have allowed imaging of the radio emission
at unprecedented detail. We present a new radio image at 44 GHz of the
supernova remnant (SNR), derived from observations performed with the Australia
Telescope Compact Array (ATCA) in 2011. The diffraction-limited image has a
resolution of $349\times225$ mas, which is the highest achieved to date in
high-dynamic range images of the SNR. We also present a new image at 18 GHz,
also derived from ATCA observations performed in 2011, which is super-resolved
to $0./!/!^{\prime\prime}25$. The new 44 and 18 GHz images yield the first
high-resolution spectral index map of the remnant. The comparison of the 44 GHz
image with contemporaneous X-ray and H$\alpha$ observations allows further
investigations of the nature of the remnant asymmetry and sheds more light into
the progenitor hypotheses and SN explosion. In light of simple free-free
absorption models, we discuss the likelihood of detecting at 44 GHz the
possible emission originating from a pulsar wind nebula (PWN) or a compact
source in the centre of the remnant.
