Whether a physical theory can be consistent without them is an interesting question I agree, it is interesting, and the answer is yes, a physical theory can be

Whether a physical theory can be consistent without them is an interesting question I agree, it is interesting, and the answer is yes, a physical theory can be completely consistent while never using complex numbers. If you don't want to use imaginary numbers, you can avoid them for almost anything by using matrices instead. Use 2x2 matrices instead of complex numbers, and instead of 1, use the identity matrix: 1 0 0 1 Instead of i, use a matrix that corresponds to a 90 degree rotation: 0 -1 1 0 This means that i * i = -1, and so pretty much everything else will just work the same way as it does with complex numbers. Adding, multiplying, calculus, all the same. No need for complex numbers if you don't like them. I do like complex numbers though. They are just a bit more concise and convenient than using matrices.