  In this paper we explore by means of the method of Lagrangian descriptors the
Julia sets arising from complex maps, and we analyze their underlying dynamics.
In particular, we take a look at two classical examples: the quadratic mapping
$z_{n+1} = z^2_n + c$, and the maps generated by applying Newton's method to
find the roots of complex polynomials. To achieve this goal, we provide an
extension of this scalar diagnostic that is capable of revealing the phase
space of open maps in the complex plane, allowing us to avoid potential issues
of orbits escaping to infinity at an increasing rate. The simple idea is to
compute the p-norm version of Lagrangian descriptors, not for the points on the
complex plane, but for their projections on the Riemann sphere in the extended
complex plane. We demonstrate with several examples that this technique
successfully reveals the rich and intricate dynamical features of Julia sets
and their fractal structure.
