I don't see why imaginary numbers were necessarily so difficult among top mathematicians back then. From pleano's axioms, we can derive the fact that any negative natural number times another negative natural number must be positive. Then this result extends to the reals, using theorems derived from those axioms as well. ( logic is a+(-a)=0 and multiply both sides by -b, then add ab to both sides) All this means that that no two real negative numbers can multiply to be positive. But surely it could be imagined that some mathematical structure times itself produce -1. It is not logically contractictory with the theorems of real numbers. So if it didn't pose any contradictions with previous mathematical work, what was the philosophical hurdle? If I make up a mathematical structure that is self consistent, then it works, no matter what it is.