  We present new imaging observations of the remnant of Supernova (SN) 1987A at
44 GHz, performed in 2011 with the Australia Telescope Compact Array (ATCA).
The $0\farcs35\times0\farcs23$ resolution of the diffraction-limited image is
the highest achieved to date in high-dynamic range. We also present a new ATCA
image at 18 GHz derived from 2011 observations, which is super-resolved to
$0\farcs25$. The flux density is 40$\pm$2 mJy at 44 GHz and 81$\pm$6 mJy at 18
GHz. At both frequencies, the remnant exhibits a ring-like emission with two
prominent lobes, and an east-west brightness asymmetry that peaks on the
eastern lobe. A central feature of fainter emission appears at 44 GHz. A
comparison with previous ATCA observations at 18 and 36 GHz highlights higher
expansion velocities of the remnant eastern side. The 18-44 GHz spectral index
is $\alpha=-0.80$ ($S_{\nu}\propto\nu^{\alpha}$). The spectral index map
suggests slightly steeper values at the brightest sites on the eastern lobe,
whereas flatter values are associated with the inner regions. The remnant
morphology at 44 GHz generally matches the structure seen with contemporaneous
X-ray and H$\alpha$ observations. Unlike the H$\alpha$ emission, both the radio
and X-ray emission peaks on the eastern lobe. The regions of flatter spectral
index align and partially overlap with the optically-visible ejecta. Simple
free-free absorption models suggest that emission from a pulsar wind nebula or
a compact source inside the remnant may now be detectable at high frequencies,
or at low frequencies if there are holes in the ionised component of the
ejecta.
