We employ the theory of topological phase transitions, of the Berezinskiĭ-Kosterlitz-Thouless (BKT) type, in order to investigate orientational ordering in four spatial dimensions that is characterized by a quaternion n−vector (i.e., n = 4) order parameter. Due to the dimensionality of the quaternion n−vector order parameter, the development of orientational order for systems that exist in four spatial dimensions must be viewed within the context of ordering in restricted dimensions. At finite temperatures, despite the development of a well-defined amplitude of the n−vector order parameter within separate regions, ordered systems that exist in restricted dimensions are prevented from developing global orientational phase-coherence as a consequence of misorientational fluctuations throughout the system. In four dimensions, this gas of misorientational fluctuations takes the form of spontaneously generated topological point defects that belong to the third homotopy group. These topological point defects must become topologically ordered in order to obtain a ground state of aligned order parameters at zero Kelvin; we argue that this topological transition belongs to the BKT universality class. We use standard ‘Metropolis’ Monte Carlo simulations to estimate the thermodynamic response functions, susceptibility and heat capacity, in the vicinity of the transition towards the ground state of perfectly aligned order parameters in our four-dimensional quaternion n−vector ordered model system. On lowering the temperature below a critical value, we identify a transition that results from the minimization of misorientations in the scalar phase angles throughout the system. The thermodynamic response functions obtained by Monte Carlo simulations show characteristic behavior of a topological ordering phase transition.