What views of the potential pools does the house's windows/sliders provide? That might influence my decision.

Views of Daisy Mountain in the North Phoenix desert

You will need a pretty substantial hedge to keep dust out of the pool area that is that close to an arena in the desert.

I don't know what grows well there, but make sure whatever you grow it is safe for the horses to eat, but not something they love.

Pretty location!

I recommend an electric cover. You'll save water, it'll stay MUCH much cleaner, and you won't need a safety fence. But covers aren't that attractive, so you don't want it to be the main view out the living room window...

I agree with Fori. You will never be able to keep the pool clean in your dusty environment without an automatic safety cover. We had an automatic pool cover installed when we remodeled our pool 10 years ago (after not having such a cover for almost 30 years) and it was one of the best decisions we have ever made. We leave the cover closed except when we are actually swimming and as a result the pool is always sparkling clean when we open it to swim or to "show it off" for guests. We save on chemicals and never have to add algaecide because algae never grows (the water is not exposed to light which is what causes algae to grow). The water only gets chlorine for sanitation and is very clean-feeling. We save on water because there is virtually no evaporation and we save on electricity because we don't have to run the pump and pool sweep very much in order to keep the pool clean. You will, however, have to have a rectangular or square pool because the cover runs on tracks under the coping and they must be straight and parallel. It is possible to design the pool and coping such that the mechanism and cover roller is hidden under the coping at one end of the pool, and so the pool looks attractive even with the cover closed. Any disadvantages? None, in my opinion. Yes, they are expensive, but the savings on water, chemicals, electricity, and your time that you would spend vacuuming and cleaning it more than compensates for the cost of the cover.