  This paper is devoted to geodesic completeness of left-invariant metrics for
real and complex Lie groups. We start by establishing the Euler-Arnold
formalism in the holomorphic setting. We study the real Lie group
$\mathrm{SL}(2, \mathbb{R})$ and reobtain the known characterization of
geodesic completeness and, in addition, present a detailed study where we
investigate the maximum domain of definition of every single geodesic for every
possible metric. We investigate completeness and semicompleteness of the
complex geodesic flow for left-invariant holomorphic metrics and, in
particular, establish a full classification for the Lie group $\mathrm{SL}(2,
\mathbb{C})$.
