Hyperbola-circle connection

I'm not sure if this is the right rubric for this question.

I know that the hyperbola and circle are both conic sections, and that one can define both in terms of foci, where the circle has one inside the circle, and the hyperbola has one outside the circle. However, I read another explanation, but I have my doubts about it. It defined the hyperbola as an "imaginary circle" in that it took the unit circle x^2 + y^2 =1, and substituted iy for y, givingx^2 + (iy)^2 =1, which is x^2 - y^2 =1, and voila, a hyperbola. Is this just someone's off the wall doodling, or is there any meaning to this?

Thanks in advance.