A complete and pure sample of quasars with accurate redshifts is crucial for quasar studies and cosmology. In this paper, we present CatNorth, an improved Gaia DR3 quasar candidate catalog with more than 1.5 million sources in the 3π sky built with data from Gaia, Pan-STARRS1, and CatWISE2020. The XGBoost algorithm is used to reclassify the original Gaia DR3 quasar candidates as stars, galaxies, and quasars. To construct training/validation datasets for the classification, we carefully built two different master stellar samples in addition to the spectroscopic galaxy and quasar samples. An ensemble classification model is obtained by averaging two XGBoost classifiers trained with different master stellar samples. Using a probability threshold of p_QSO_mean > 0.95 in our ensemble classification model and an additional cut on the logarithmic probability density of zero proper motion, we retrieved 1,545,514 reliable quasar candidates from the parent Gaia DR3 quasar candidate catalog. We provide photometric redshifts for all candidates with an ensemble regression model. For a subset of 89,100 candidates, accurate spectroscopic redshifts are estimated with the Convolutional Neural Network from the Gaia BP/RP spectra. The CatNorth catalog has a high purity of > 90% while maintaining high completeness, which is an ideal sample to understand the quasar population and its statistical properties. The CatNorth catalog is used as the main source of input catalog for the LAMOST phase III quasar survey, which is expected to build a highly complete sample of bright quasars with i < 19.5.