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 jets 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.
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