In Chapter 434 we see Shanks pay a visit to Whitebeard. As he walks toward him activating his Conqueror’s Haki, parts of the Moby Dick crack out of the pressure of his power. In this case, Shanks was just trying to appear slightly intimidating. In a more serious situation, it’s not crazy to assume he could use this same Haki to affect something bigger, like the air pressure, therefore creating storms.
Later in the same chapter, when Shanks and Whitebeard (two characters with confirmed Conqueror’s Haki) clash, the sky splits in two. This could very well be a side effect of this technique of Conqueror’s Haki when it’s used. This is the first time we see this happen in the story, but not the last.
In Chapter 951 we see another clash between two Emperors, Kaido and Big Mom. The effects are similar to the one seen previously, with the sky splitting, and the sea raging around Onigashima.
In the most recent clash in Chapter 966 between Roger and Whitebeard, we see a great gust of wind that pushes back the clouds, and then a storm in the island during their 4 days fight.
I believe that’s all of the evidence there is regarding this potential type of Advanced Conqueror’s Haki. If this type of ability actually exists, out of the 14 characters that currently confirmed to have Conqueror’s Haki, there would be 5 confirmed to have it: Roger, Whitebeard, Kaido, Big Mom and Shanks (and Dragon, assuming he’s a Haki user). That would make it even more rare than regular Conqueror’s Haki.
In summary, an Advanced form of Conqueror’s Haki is creating storms that can be controlled by the user to summon wind, lighting, and dominate the sea.
*Theory by RoderickThe13