In this work, we focus on a class of timelike rotational surfaces in Minkowski space $\Bbb E^4_1$ with 2-dimensional axis. There are three types of rotational surfaces with 2-dimensional axis, called rotational surfaces of elliptic, hyperbolic or parabolic type. We obtain all flat timelike rotational surface of elliptic and hyperbolic types with pointwise 1-type Gauss map of the first and second kind. We also prove that there exists no flat timelike rotational surface of parabolic type in $\Bbb E^4_1$ with pointwise 1-type Gauss map.