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 4237.3deg^2^ of the Sloan Digital Sky Survey footprint. These observations conclude the ELQS and we present the properties of the full ELQS quasar catalog, containing 407 quasars over 11838.5deg^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 ~36% (109/298) in the known population at these redshifts and magnitudes, while we further are able to retain a selection efficiency of ~80%. On the basis of 166 quasars from the full ELQS quasar catalog, which adhere to the uniform criteria of the Two Micron All Sky Survey 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}~-4.1, and we can constrain the bright-end slope to {beta}<=-3.4 with 99% confidence. The density is well modeled by the exponential redshift evolution, resulting in a moderate decrease with redshift ({gamma}~-0.4).