The human memory works in funny ways. Yesterday, I thought of my first semester maths tutor, D. I really had a crush on him, awful. Some day, we happened to be alone together in the elevator. I had 25 floors to make a good impression.
We were just discussing complex numbers in maths. Parallel to this, the theoretical physicists had the harmonic oscillator on their schedule. And well, you know how it goes: plug in an exponential, find the complex solutions, take the real part. Sure, I could solve these equations, but I didn't understand where the imaginary part goes. If that question makes any sense. Plus I had learned Special Relativity with ict, which added to my confusion .
So, I asked D. why only the real numbers 'exist' and where the others are. (That means I must have found that a really good question.) 25 floors he had no way to get out of this.
This question came back to me yesterday when I read through your comments to The Mathematical Universe. See, as far as I know nobody has ever measured an observable to be a complex number. So whatever 'mathematical structure' constitutes the 'external physical reality' of our universe, complex numbers don't seem to be part of it .
However, we know that there are problems which can't be solved purely within the real numbers, say, take a square root of the Klein-Gorden equation. So the universe might try to evolve an initially real valued state, it wants to become complex, but the complex numbers aren't part of our 'external physical reality'. Then what? Do we get a cosmic error message? Does the unintelligent designer of our local patch in the multiverse get an F and fails the exam? Does the wave-function jump into another universe where the complex numbers exist, and then collapses back into ours?
The next time D. and I ran into danger of sharing the elevator, he had to use the bathroom really urgently. Gee, my whole life could have been different, if it wasn't for these complex numbers.
 Still today, Wick-rotations seem like magic to me. Is there any good reason why that works?
 Since we are dealing with complex numbers every day (well, some of us) this then means human thoughts are not real?