For hyperbolic equations, characteristics tell how a signal propagates. For example, the equation $u_{tt} - c^2 u_{xx} = 0$ has two characteristics $x\pm c t$. This means a signal propagates at speeds $\pm c$ starting from point $x$.
A similar, but elliptic equation $u_{tt} + c^2 u_{xx} = 0$ has two characteristics $x \pm i c t$. Now things are imaginary. But it must still mean something, right? For example, $u_{tt} + x^2 u_{xx} = 0$ is also elliptic but with two different characteristics. Can we extract some useful information of their (different) elliptic behavior, by looking at the imaginary characteristics?
In the first case the general solution is $$u=F(x-ct)+G(x+ct)$$ (method of characteristics)
In the second case the general solution is $$u=F(x-ict)+G(x+ict)$$ (method of complex variables). Sophisticated versions of this have been used to solve problems of transonic airflow.