  The focus of this paper is on the distribution function of the rightmost
eigenvalue for the complex elliptic Ginibre ensemble in the limit of weak
non-Hermiticity. We show how the limiting distribution function can be
expressed in terms of an integro-differential Painlev\'e-II function and how
the same captures the non-trivial transition between Poisson and Airy point
process extreme value statistics as the degree of non-Hermiticity decreases.
Our most explicit new results concern the tail asymptotics of the limiting
distribution function. For the right tail we compute the leading order
asymptotics uniformly in the degree of non-Hermiticity, for the left tail we
compute it close to Hermiticity.
