  Despite the fact that kpc-scale inverse-Compton (iC) scattering of cosmic
microwave background (CMB) photons into the X-ray band is mandated, proof of
detection in resolved quasar jets is often insecure. High redshift provides
favourable conditions due to the increased energy density of the CMB, and it
allows constraints to be placed on the radio synchrotron-emitting electron
component at high energies that are otherwise inaccessible. We present new
X-ray, optical and radio results from Chandra, HST and the VLA for the core and
resolved jet in the $z=3.69$ quasar PKS J1421-0643. The X-ray jet extends for
about $4.5''$ (32 kpc projected length). The jet's radio spectrum is abnormally
steep and consistent with electrons being accelerated to a maximum Lorentz
factor of about 5000. Results argue in favour of the detection of
inverse-Compton X-rays for modest magnetic field strength of a few nT, Doppler
factor of about 4, and viewing angle of about $15^\circ$, and predict the jet
to be largely invisible in most other spectral bands including the far- and
mid-infrared and high-energy gamma-ray. The jet power is estimated to be about
$3 \times 10^{46}$ erg s$^{-1}$ which is of order a tenth of the quasar
bolometric power, for an electron--positron jet. The jet radiative power is
only about 0.07 per cent of the jet power, with a smaller radiated power ratio
if the jet contains heavy particles, so most of the jet power is available for
heating the intergalactic medium.
