Level sets of certain classes of $\alpha$-analytic functions

For an open set $V\subset {\mathbb{C}}^n$, denote by ${\mathcal{M}}_{\alpha }(V)$ the family of $\alpha$-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded domain $\mathrm{\Omega }\subset {\mathbb{C}}^n$, with continuous boundary (that in each variable separately allows a solution to the Dirichlet problem), a function $f \in \mathscr{M}{\alpha}(\Omega\setminus f^{-1}(0))$automaticallysatisfies$f\in \mathscr{M}{\alpha}(\Omega)$, if it is $C^{{\alpha }_j-1}$-smooth, in the $z_j$ variable, $\alpha \in {\mathbb{Z}}_{+}^n$, up to the boundary. For a submanifold $U\subset {\mathbb{C}}^n$, denote by ${\mathfrak{M}}_{\alpha }(U)$ the set of functions locally approximable by $\alpha$-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a $C^3$-smooth hypersurface, $\mathrm{\Omega }$, a member of ${\mathfrak{M}}_{\alpha }(\mathrm{\Omega })$, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form.