  We have designed the Extremely Luminous Quasar Survey (ELQS) to provide a
highly complete census of unobscured UV-bright quasars during the cosmic noon,
$z=2.8-5.0$. Here we report the discovery of 70 new quasars in the ELQS South
Galactic Cap (ELQS-S) quasar sample, doubling the number of known extremely
luminous quasars in $4,237.3\,\rm{deg}^2$ of the SDSS footprint. These
observations conclude the ELQS and we present the properties of the full ELQS
quasar catalog, containing 407 quasars over $11,838.5\,\rm{deg}^2$. Our novel
ELQS quasar selection strategy resulted in unprecedented completeness at the
bright end and allowed us to discover 109 new quasars in total. This marks an
increase of $\sim36\%$ (109/298) to the known population at these redshifts and
magnitudes, while we further are able to retain a selection efficiency of
$\sim80\%$. On the basis of 166 quasars from the full ELQS quasar catalog, who
adhere to the uniform criteria of the 2MASS point source catalog, we measure
the bright-end quasar luminosity function (QLF) and extend it one magnitude
brighter than previous studies. Assuming a single power law with exponential
density evolution for the functional form of the QLF, we retrieve the best fit
parameters from a maximum likelihood analysis. We find a steep bright-end slope
of $\beta\approx-4.1$ and we can constrain the bright-end slope to
$\beta\leq-3.4$ with $99\%$ confidence. The density is well modeled by the
exponential redshift evolution, resulting in a moderate decrease with redshift
($\gamma\approx-0.4$).
