It's an interesting and unresolved question whether imaginary numbers are fundamental to the nature of physics or just a handy tool to calculate stuff. They are certainly handy for calculating - they are an easy way to represent oscillations.
Maybe it's a mistake to allow imaginary to ever be separate from real numbers ? Another post here brought up how easily it is to work with imaginaries if you just treat them as the vector [re im]. Maybe (philosophically speaking), there are no purely real numbers. Could it be that all quantities in nature must contain a (sometimes zero) Im component ? That might be a more satisfying interpretation than just allowing them to creep in when they are absolutely required, such as polynomial equations.