In the case of the Kerr metric, for a high enough angular velocity such that on transformation to Boyer-Lindquist coordinates yields complex coordinates for the event horizon, why is it assumed then that the coordinates "disappear"? are complex coordinates completely outlawed in physics?
What about these coordinates means they're designated as non-existent?
are complex coordinates completely outlawed in physics?
Complex coordinates are “outlawed” specifically in general relativity, which is the relevant physics for your question.
The mathematical framework of general relativity is based on pseudo-Riemannian manifolds. These are defined as structures that are locally isomorphic to $\mathbb{R}^n$. So they cannot be complex by definition of the framework itself.
Complex coordinates can be used in other scenarios. And they were considered in the early days of special relativity. However, the pseudo-Riemannian approach can represent the early complex coordinate metric of special relativity as well as many other metrics that the complex one cannot. So there was no advantage to that approach over the current one.
Of course, it would be possible to relax the requirement and/or consider manifolds locally isomorphic to $\mathbb{C}^n$ and see where the math leads. But just because the modified math would allow it doesn’t mean that it will describe any physical system.
Many physical theories use complex numbers. Quantum mechanics is the obvious example, but even everyday theories like circuit theory use complex numbers, and then we have the more esoteric theories like twistor theory where the space in which it is formulated has three complex dimensions.
So there is nothing wrong with a theory using properties that are complex, but at the end of the day a theory has to predict observable quantities and millennia of experiments by physicists suggests that observable quantities are always real numbers.
In GR the observable quantities are generally Lorentz scalars like the proper time/length, and experiment suggests these have to be real. The problem with naively using a complex dimension is that it allows the proper time to be complex and this contradicts experiment.
Quantum mechanics gets away with it because the wave function is not an observable and the observable quantities are always real because of the way the theory is constructed. Circuit theory gets away with it because it doesn't actually use complex numbers - the complex number is just a way of encoding magnitude and phase into a single quantity. I have no idea how twistor theory gets away with it but apparently it does!
Complex numbers are used in describing electromagnetic radiation to represent the interaction between the electrical field and magnetic fields as they propagate over time.
All we ask of math in physics generally is that it accurately model the universe. The fact that calling one of these axes imaginary makes the math work does not mean it is imaginary in the colloquial sense, it just means that the square root of -1 happens to be a more useful value than was expected when it was named.
It is probably true that a quantity becoming complex doesn't always or necessarily imply something unphysical. But it does not seem worth the space to give further justification, in most cases.
It could be that two different quantities become complex, but only their product is physically measurable. For example, Taylor & Wheeler make a local Lorentz boost from a static frame to a falling "raindrop" frame in Schwarzschild spacetime (2000, Exploring black holes, §B4, box titled "Metric for the Rain Frame"). However, inside the horizon, this so-called boost is actually superluminal, so the Lorentz factor $\gamma$ becomes complex. Yet, the terms combine so everything turns out real in the end. As I recall there is something similar in Misner, Thorne & Wheeler. Admittedly, the derivation could probably be fixed to avoid complex numbers, so this may not be the best example.
The Newman-Penrose formalism for general relativity uses complex vectors.
Electromagnetism is conveniently formulated using complex vectors. However this is only one of many ways to formulate EM.
An unconventional view is offered by "geometric algebra", which is essentially another name for Clifford algebra. This community looks for physical or geometric interpretations to replace complex numbers. For example, you can define the square of a plane (i.e. a plane multiplied by itself). In some cases, the result is -1. But this avoids scalars $\mathbb C$. Instead, complex numbers are associated with rotation within a plane.
