Suppose you have the following differential equation:
$au_{xx}+ 2bu_{xy} + cu_{yy} = 0$.
For real $a,b,c$ the classification into elliptical, hyperbolic, or parabolic is based on calculating the sign of the determinant $b^2 - ac$. Furthermore, I can divide all the coefficients by $a$ and have the same sign for $b^2 - ac$; that is, the differential equation
$u_{xx}+ \frac{2b}{a}u_{xy} + \frac{c}{a}u_{yy} = 0$.
has the same classification because the determinant is $\frac{1}{a^2}(b^2-ac)$.
What confuses me is the following: suppose $a$ is imaginary (e.g., $a=i$), while $b$ and $c$ are real. If I try calculating $b^2 - ac$, I end up with an imaginary number. Furthermore, if I divide all the coefficients by $a$, suddenly the sign of the determinant changes, meaning that the two equations above have different classifications.
Is it possible to classify a PDE as hyperbolic, parabolic, or elliptical with complex coefficients?