It turns out that the "Catalinas" are just one mountain range in a belt that sweeps from SE to NW across the state (and into California). Furthermore, the Arizona complexes are part of a bigger group that run from Mexico to Canada. And they are not found only in western north America, there are core complexes in Greece and who knows where else.
Here are two maps showing their distribution in Arizona and the Western US.
So what is a core complex? We can cite the Santa Catalina - Rincon complex as an example, and that is fine if you (like me) are closely familiar with those mountains and the rocks exposed in them. You may sometimes hear them called "Gneiss domes", but this terminology is deprecated and rarely used.
Core complexes are enigmatic. The only began to be recognized around 1975 when geologists realized that the metamorphic and igneous rocks in such mountains were not ancient Precambrian rock -- they are actually quite young (mid Cenozoic). Attention began to be focused on them beginning around 1980 and they have gotten more and more attention from geologists over the subsequent 50 years.
I would go to South Mountain in Phoenix, after reading what Steve Reynolds has written about it. The Whipples are a must. Snake Range is historically important and still controversial. The Raft Rivers give a sense for the core complexes way up north.
Some recommended authors:
George H. Davis Gary Axen (detachment faulting) Jon Spencer Gordon Lister Greg Davis, Steve Reynolds Art Snoke John Platt Peter Coney ...
Tom's Geology Info / tom@mmto.org