A complex solution can be valid but you never measure a complex number. I'd argue that in situations like using complex numbers to simulate time varying electrical activity the ontological status of the imaginary part of the solution is uncontroversial: the complex numbers in that situation have no ontological status at all and what is present is charges. In that case we're simply using the complex numbers as a convenient notation for a variable and its conjugate. In quantum mechanics one is more easily led to wonder about whether the ontological status of the complex numbers in that theory really can be settled so easily.

>A complex solution can be valid but you never measure a complex number.

I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.

Nothing stops you from representing the value that way, but when you go to a meter or lay a yardstick against something, you are measuring a real number (or, at the very least, a number which has no complex character to it). "This many ticks on a ruler" or "this many clicks on a clock."

Well, for one thing, for such quantity to make physical sense, both the real part and the imaginary part should be of the same dimension, e.g. "length." Also, the result of a measurement is supposed to come from (be an eigenvalue of) an observable - an operator, and, on the one hand, I think I'd have a hard time conjuring one up; on the other hand, the eigenvalues are "supposed to be" real anyway! So, no, that doesn't work.