Bighorn is private, so first you have to gain entry through a member. As to the course, there are actually two, Canyons and Mountain. The Mountain is the original and hosted the Skins Game a few times in the mid 90's - VERY hilly, hence the name. Canyons opened in 1998 and hosted the Battle at Bighorn with Tiger several years, and recently four years of the Ladies Samsung World Championship. I assume that's the one you saw. Very good track, one of the top ten in the desert, and if you like Fazio, will definitely like it.
Many of the best in desert are private, here are a few I'd have on my list. Tradition (my personal favorite, a Palmer design), StoneEagle (Doak design, great vistas, DEVILISH greens), The Quarry, Palms, Madison Club would be on the private list with Bighorn.
On the public list, Classic Club (host of the Hope), PGA West, La Quinta Mountain a few of the best.