We describe the first non-classical 4-dimensional Minkowski planes and show that they have 6-dimensional automorphism groups. These planes are obtained by a construction of Schroth [18] from generalized quadrangles associated with the semiclassical 4-dimensional Laguerre planes. All 4-dimensional Minkowski planes that can be obtained in this way from the semiclassical 4-dimensional Laguerre planes are determined.