The theory of polynomial bases in one complex variable, as given by Whittaker and Cannon, had been extended partially to the scope of Clifford analysis, where many results have been dealt with from different aspects. The aim of this paper is to introduce a certain type of bases of axially monogenic polynomials generated from the maximum modulus base. Determination of the convergence properties of such bases are closely related to the growth behavior of associated entire axially monogenic functions. We start by investigating the presented bases of monogenic polynomials associated with entire axially monogenic functions in the way we are going to indicate, and then show how this leads us to certain relations involving maximum moduli of an entire axially monogenic function on a sequence of balls with increasing radii. In this concern, we point out that Hadamard's three hyper-balls theorem can be employed to justify some of our results. The significance of this study is due to the variety of the results which are not normally expected.