  In this work, we use the concept of quaternion time and demonstrate that it
can be applied for description of four-dimensional space-time intervals. We
demonstrate that the quaternion time interval together with the finite speed of
light propagation allow for a simple intuitive understanding of the time
interval measurement during arbitrary relative motion between a signal source
and observer. We derive a quaternion form of Lorentz time dilation and show
that the norm corresponds to the traditional expression of the Lorentz
transformation and represents the measured value of time intervals, making the
new theory inseparable from the theory of measurement. We determine that the
space-time interval in the observer reference frame is given by a conjugate
quaternion expression, which is essential for proper definition of the
quaternion derivatives in the observer reference frame. Then, we apply
quaternion differentiation to an arbitrary potential, which leads to
generalized Lorentz force. The second quaternion derivative of the potential
leads to expressions similar to generalized Maxwell equations. Finally, we
apply the resulting formalism to electromagnetic and gravitational interactions
and show that the new expressions are similar to the traditional expressions,
with the exception of additional terms, related to scalar fields, that need
further study and experimental verification. Therefore, the new mathematical
approach based on Hamilton's quaternions may serve as a useful foundation of
the unified theory of space-time and matter.
