We present the Omega Number System, an extension of the complex number system that integrates both infinitary and infinitesimal scales into a unified, hierarchical framework. Our construction is anchored by a fundamental scaling element, \(\Omega\), rigorously defined as the hyperreal corresponding to the equivalence class of the standard sequence \((1,2,3,\dots)\) via the ultrapower construction. Central to our approach is the lifting function \(L^\Omega(n)\), which elevates \(\Omega\) into a graded hierarchy; wherein the index \(n=0\) corresponds to the classical (finite) complex domain, \(n>0\) to increasingly large (infinitary) magnitudes, and \(n<0\) to infinitesimal values. From \(\Omega\) and \(L^\Omega(n)\), we derive key foundational objects, including the absolute zero \(\underline{0}\), the almost zero \(\overline{0}\) (capturing the continuum of infinitesimal values), and its distinguished member, the canonical zero \(0^* = \Omega^{-1}\); together with the identity \(1\), these objects extend classical arithmetic in a coherent manner. We illustrate our approach through a foundational base linear model that extends familiar arithmetic while capturing phenomena beyond classical constructs at each index level, yielding unique hierarchical expansions in which infinitesimals and infinite magnitudes coexist systematically. We also discuss potential applications, such as the reinterpretation of classical singularities and the regularization of divergent behaviors. Although some aspects—such as multivalued or probabilistic interpretations of certain functions—remain exploratory, the Omega Number System provides a flexible foundation for further analytical developments. Future work will pursue a more complete axiomatic foundation, abstract algebraic generalizations, and connections to advanced problems in pure mathematics and theoretical physics, thereby laying the groundwork for Omega Analysis—an extension of classical complex analysis into the transfinite realm.