As we all know the hyperbolic PDE $$\partial_x^2 u - \partial_y^2u=0$$ has the general solution $u(x,y)=f(x\pm y)$ for any $f$ twice differentiable.
This comes from the fact that we can trivially "factor" the differential operator $\partial_x^2 - \partial_y^2 = (\partial_x + \partial_y)(\partial_x - \partial_y)$.
When we look at the elliptic PDE $$\partial_x^2 u + \partial_y^2u=0$$ we have the problem that we cannot factor the operator $\partial_x^2 + \partial_y^2$ in the same way.
Then I realised that if we work in the analytic category, we can factor this operator, $\partial_x^2 + \partial_y^2 = (\partial_x + i\partial_y)(\partial_x - i\partial_y)$.
This means that if $f(x\pm y)$ solves the hyperbolic PDE, then $f(x\pm iy)$ solves the elliptic PDE. So in a way, the "elliptic behaviour" coexists with the "hyperbolic behaviour" once we look in $\mathbb C$.
I would be interested to learn what is known about this relation. Is there any good source to read about such things?