Depends on what the CSA gets in New Mexico/Arizona. OTL they lost big time in that campaign, and the US was in position to send more forces to the Southwest had they won (although this would give the Plains Indians even more free reign than they already had). The Southwest is not an area which the CSA is likely to keep in any realistic CSA victory.
But assuming they did keep New Mexico, they now have to go deal with the Apache and the Comanche. The Comanche could sign a treaty with the United States, keeping them out of US territory in exchange for continuing their attacks on Confederate Texas and New Mexico. It's a war the CSA is likely to win, even with the US supplying the Comanche with weapons, ammo, and food, but one which will be very costly. As for the Apache, the US also will likely be able to supply them, and the same goes for the Navajo, who will also be an utter headache to the CSA. The Confederate Indian Wars will be a challenging and expensive affair, even though the sheer numbers of CSA soldiers and population will grind the American Indians down over the years.
I think it's also likely the CSA will try and grab a Pacific port (this was the goal of the New Mexico campaign, but the CSA conquering California is utterly unrealistic). Guaymas is probably the best port since it's more secure from the US than anything in Baja California, but would also be more difficult to get Mexico to part with it. Taking most of Sonora adds to the headaches the CSA will have with American Indians, since the Yaqui Wars was the last Indian War fought, and the Yaquis were large in number and likely to be armed by both Mexico and the US.
The United States is likely to have a similar experience in the West as OTL, though maybe a bit more peaceful. For instance, the Sand Creek massacre won't happen since the soldiers who attacked the Cheyenne will instead be fighting with the CSA. It's more likely to be militarised earlier since it's a border region, so the Indian Wars will likely end a lot sooner. Cross border trade will lead to a lot of smuggling and illegal operations which will become part of Western lore. Slaves will likely try to escape to the Western US (or Mexico as OTL). A Confederate Camel Corps would also be pretty interesting to see too, since after all, the original Camel Corps was Jefferson Davis's idea. Such a thing could lead to the US also re-establishing a Camel Corps, leading to more use of camels in the west.
Overall, pretty similar to OTL but with some striking differences that would make TTL's westerns rather unique.