This paper concerns 4-dimensional ( topological locally compact connected) elation Laguerre planes that admit large automorphism groups. In particular, it is shown that such a plane is classical if its automorphism group is at least 11- dimensional. Furthermore, the elation Laguerre planes admitting simple Lie groups of automorphisms are investigated and various characterizations of the classical complex Laguerre plane and the semi-classical Laguerre planes are obtained.