  In this article, we define amorphic complexity for actions of locally compact
$\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity,
originally introduced for $\mathbb Z$-actions, is a topological invariant which
measures the complexity of dynamical systems in the regime of zero entropy. We
show that it is tailor-made to study strictly ergodic group actions with
discrete spectrum and continuous eigenfunctions. This class of actions
includes, in particular, Delone dynamical systems related to regular model sets
obtained via Meyer's cut and project method. We provide sharp upper bounds on
amorphic complexity of such systems. In doing so, we observe an intimate
relationship between amorphic complexity and fractal geometry.
