The title pretty much captures my question. I understand that there are both real and complex $K$-theories of Hausdorff topological spaces depending on whether we look at real or complex bundles over the space. However, my understanding is that in a lot of contexts, when considering whether to look at things defined over $\mathbb{R}$ or $\mathbb{C}$, usually $\mathbb{C}$ is the better choice. I understand that both theories have Bott periodicity (though the real theory has period $8$ and the complex theory has period $2$). Is one of the theories (i.e. real or complex) in some sense "stronger" than the other, or just preferable over the other for whatever reason?
My apologies if these are very basic questions. I'm only starting to learn about it, but I find it interesting that most texts when introducing the topological $K$-theory introduce both, but don't seem to recommend one over the other. I also find it interesting that the C*-algebraic $K$-theory generalizes the complex theory, but none of the texts I've found discuss a generalization of the real theory, at least not when introducing the $K$-groups and functors.
Thanks!
EDIT: Eric Wofsey pointed out that the "stronger" theory isn't necessarily the "preferable" theory, as the stronger one is probably harder to compute; I'm interested in both questions. My assumption would be that if the real and complex theories were equivalent, then the complex theory might win out over the real one if only because it's nicer to work with 6-term cyclic exact sequences than 24-term cyclic exact sequences.
Also, I know that there is a $K$-theory of real C*-algebras, but I also had to seek out this fact, as it wasn't found in any introductory texts I could get my hands on.