Its botanical name is Fallugia paradoxa. This is a native southwest shrub and I planted it in a couple of places in the yard, mostly because it is one of the few shrubs that is available at Tucson plant nurseries. By the way, the popular name refers to the seed head which resembles an Indian headdress.
I wonder what the "paradoxa" refers to? Pro: it is an easily grown plant that looks good in the landscape, has flowers and an interesting seedhead? Con: it produces an incredible amount of seeds that take hold close by and the seedlings are difficult to get rid of? Don't get me wrong, I like Apache plume, but the carpet of seeds below it right now is a bit daunting. Mom is so very generous!
Here is the plant in our driveway circle, its bloom and seed head, and all those seeds just waiting to be carried on our Spring winds and sprout somewhere else.
A little bonus: a blooming Santa Rita cactus right next to it, also in the driveway circle.