In a particular example of a real conic $ w=f(z)=z^2 $ the graph is a parabola along real $w,z $ axes. When $w,z$ are complex the real and imaginary parts separately graph to a set of real orthogonal conics.
When $w=f(z) = u+i v $ is a conic of complex coefficients how does each of the (orthogonal) resultant real component conics set relate to the original real conic before complexifcation?
How do the eccentricity, rotation and translation of the origin of the first real graph relate to each of real/imaginary parts of the new function and its graph?
EDIT1:
Some plots with real/imaginary coefficients in what appears to be a simple interaction .
