In an earlier paper a q-Schrödinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q-deformation of the Witt algebra. This q-deformation is a deformation in the category of Lie algebras and hence the corresponding q-Witt algebra has a trivial Hopf algebra structure. In this paper, we extend the above algebra by the addition of a set of shift-type generators, which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics. The latter gives rise to a new kind of evolution equation and it is shown that in the limit q1 a specific class of Schrödinger equations is obtained from it. This specification of a particular class is a new phenomenon, because in earlier references, where a different q-deformation has been implemented or no deformation has been used at all, such a class could not be determined uniquely. The extended algebra used here has a nontrivial Hopf structure. The appearance of the shift-type generator in the q-deformed picture hence leads to a selection of a particular type of dynamics and delivers in the limit q1 new information for the characterization of the corresponding dynamics in the undeformed situation.