The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier… - John C. Baez

The conformal invariance of the Yang-Mills equations in four dimensions greatly facilitates the study of the temporal asymptotic behavior of their solutions.

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics.

The most popular approach to quantum gravity is string theory. Despite decades of hard work by many very smart people, it's far from clear that this theory is successful. It's made no predictions that have been confirmed by experiment. In fact, it's made few predictions that we have any hope of testing anytime soon! Finding certain sorts of particles at the big new particle accelerator near Geneva would count as partial confirmation, but string theory says very little about the details of what we should expect. In fact, thanks to the vast "landscape" of string theory models that researchers are uncovering, it keeps getting harder to squeeze specific predictions out of this theory.