We present results on the dynamics of standard univariate polynomials over Quaternions and Octonions from an algebraic point of view. We look at periodic and fixed points, and emphasize the need to carefully define what we mean by these terms when it comes to dynamics over non-commutative rings. For quadratic monics over the Octonions, we provide criteria for classifying fixed points as attracting, repelling or "ambivalent", generalizing the complex case. This is a joint work with Adam Chapman from The Academic College of Tel Aviv–Yaffo.