A new class of regular quaternionic functions, defined by power series in a natural fashion, has been introduced in recent years. Several results of the theory recall the classical complex analysis, whereas other results reflect the peculiarity of the quaternionic structure. A more recent paper identified a larger class of domains, on which the study of regular functions is most natural and not limited to the study of quaternionic power series. In the present paper we extend some basic results concerning the algebraic and topological properties of the zero set to regular functions defined on these domains. We then use these results to prove the Maximum and Minimum Modulus Principles and a version of the Open Mapping Theorem in this new setting.

13 pages