  In this paper we study the asymptotics (as $n\to \infty$) of the sequences of
Laguerre polynomials with varying complex parameters $\alpha$ depending on the
degree $n$. More precisely, we assume that $\alpha_n = n A_n, $ and $ \lim_n
A_n=A \in \mathbb{C}$. This study has been carried out previously only for
$\alpha_n\in \mathbb{R}$, but complex values of $A$ introduce an asymmetry that
makes the problem more difficult.
  The main ingredient of the asymptotic analysis is the right choice of the
contour of orthogonality, which requires the analysis of the global structure
of trajectories of an associated quadratic differential on the complex plane,
which may have an independent interest.
  While the weak asymptotics is obtained by reduction to the theorem of
Gonchar--Rakhmanov--Stahl, the strong asymptotic results are derived via the
non-commutative steepest descent analysis based on the Riemann-Hilbert
characterization of the Laguerre polynomials.
