I don't even understand why this is such a big contention, real and complex numbers are basically the same (equal cardinality sets), the reason we use C is because it makes math easier.
I reject the premise of the question, real numbers are fake anyway and complex numbers are equally as fake. We made both of those up. Just use the more ergonomic formalism bro.
God, we had so much fun doing this at my uni when creating an example DB table for an exam (only it was fake song/band combinations). Are you sure your coworker isn't a robot themselves?