  A newly identified kiloparsec-scale X-ray jet in the high-redshift z=3.89
quasar 1745+624 is studied with multi-frequency Very Large Array, Hubble Space
Telescope, and Chandra X-ray imaging data. This is only the third large-scale
X-ray jet beyond z>3 known and is further distinguished as being the most
luminous relativistic jet observed at any redshift, exceeding 10^45 erg/s in
both the radio and X-ray bands. Apart from the jet's extreme redshift,
luminosity, and high inferred equipartition magnetic field (in comparison to
local analogues), its basic properties such as X-ray/radio morphology and radio
polarization are similar to lower-redshift examples. Its resolved linear
structure and the convex broad-band spectral energy distributions of three
distinct knots are also a common feature among known powerful X-ray jets at
lower-redshift. Relativistically beamed inverse Compton and `non-standard'
synchrotron models have been considered to account for such excess X-ray
emission in other jets; both models are applicable to this high-redshift
example but with differing requirements for the underlying jet physical
properties, such as velocity, energetics, and electron acceleration processes.
One potentially very important distinguishing characteristic between the two
models is their strongly diverging predictions for the X-ray/radio emission
with increasing redshift. This is considered, though with the limited sample of
three z>3 jets it is apparent that future studies targeted at very
high-redshift jets are required for further elucidation of this issue. Finally,
from the broad-band jet emission we estimate the jet kinetic power to be no
less than 10^46 erg/s, which is about 10% of the Eddington luminosity
corresponding to this galaxy's central [abridged]
