In this work, we use the key assumption that division algebras play a key role in description and prediction of natural phenomena. Consequently, we use division algebra of quaternions to describe the four-dimensional space-time intervals. Then, we demonstrate that the quaternion space-time together with the finite speed of signal propagation allow for a simple, intuitive understanding of the space-time interval transformation during arbitrary motion between a signal source and observer. We derive a quaternion form of the Lorentz time dilation and suggest that it's real scalar norm is the traditional form of the Lorentz transformation, representing experimental measurements of the space-time interval. Thus, the new quaternion theory is inseparable from the experimental process. We determine that the space-time interval in the observer reference frame is given by a conjugate quaternion expression, which is essential for a proper definition of quaternion gradient operator. Then, we apply the quaternion gradient to an arbitrary quaternion potential function, which leads to the unified expressions of force fields. The second quaternion differentiation results in the unified Maxwell equations. Finally, we apply the resulting unified formalism to electromagnetic and gravitational interactions and show that the new expressions are similar to the traditional equations, with the novel terms related to scalar fields and velocity dependent components. Furthermore, we obtain two types of force fields and four types of matter density expressions, which require further theoretical and experimental study. Therefore, the new mathematical framework based on quaternion algebra and quaternion calculus may serve as the foundation for a unified theory of space-time and matter, leading to a useful enhancement of the traditional theories of special and general relativity.