In Mexico there are no monuments with his figure because he is considered a villain, but an entire sea bears his name
The first thing to understand is that for Hernán Cortés the taking of Mexico Tenochtitlán was rather the beginning and not the end of his conquests. Although having defeated such an incredible city brought him great fame, the truth is that he was still looking for the promised cities full of gold, such as El Dorado or California and above all, the route by sea to India and the Islands of the spices. In addition, Cortés had some legal problems because he had made the conquest without the permission of the Spanish authorities in the Caribbean, so he had to justify his actions to the king. In 1929 he traveled to Spain to endorse his title as governor of New Spain, a kingdom that he himself had baptized and which was now governed by Nuño Beltrán de Guzmán. The king denies him the title of governor but grants him the noble title of “Marqués del Valle de Oaxaca”. On the coast of that region, on the Isthmus of Tehuantepec, he founded the first shipyard to manufacture boats, because if he found new lands then he would be able to bear the title of governor as an individual belonging to the nobility.
Thanks to that shipyard, Hernán Cortés sponsors expeditions to continue exploring the sea, now on the Pacific side. In the first expedition, his cousin Diego Hurtado went north in search of a supposed maritime passage between the Atlantic and the Pacific, or as it was called at that time, the Ocean Sea and the South Sea, also they were looking for the mythical island ruled by Princess Calafia, called California. This first expedition had no news, so it is assumed that they were shipwrecked. In 1533 a new expedition set sail from Manzanillo with two ships, one reported having found the Revillagigedo Islands and the other which took the same north course as the first expedition suffered a mutiny, where people loyal to Cortés disembarked and the mutineers continued towards the Gulf of California, it would be known later that they arrived at the Baja California Peninsula to be killed by the natives.
Cortez leads the expedition
The next expedition is led by Cortés himself, given the failure of the previous ones, the conquistador decides to go to sea further north, in Sinaloa, which was a land controlled by Nuño Beltrán de Guzman, but he has a large army on land who challenges Nuño and he could not stop his pace. On May the 3th, 1535, Hernán Cortés arrived at the current bay of La Paz. Instead of the kingdom of Princess Calafia, he found a beautiful but unpromising land for colonization. Even so, the name of California was given to that land and he leaves some men in that land.
Five years later, Cortés would put Francisco de Ulloa in command of a fourth expedition, he toured to the Bay of La Paz and saw that the Spanish colony had disappeared. He reached the north of the sea where the Colorado River flows and began to use the name of the conqueror to refer to the great gulf he was navigating. His chronicles reach New Spain but Ulloa continues sailing to the other side of the Peninsula where no more is known about him. For many years this sea would remain unexplored but bearing the name of the mythical conqueror of Mexico.