My approach was this: okay so let $f : \mathbb{C} \to \mathbb{C}$ be defined by $f(z) = p(z)$, then note that $f'(z) = mz^{m-1} +a_1(m-1)z^{m-2} + \dots + a_{m-1}$. By the fundamental theorem of algebra $f'(z)$ has at most $m-1$ roots, call them $\eta_1, \dots \eta_{m-1}$, and $f'(n_i) = 0$ for all of these roots.

Then for all $z \in \mathbb{C} \setminus \{n_1, \dots, n_{m-1}\}$ we have $f'(z) \neq 0$ and hence $df_z : T_z\mathbb{C} \to T_z\mathbb{C}$ has rank $1$ as a $\mathbb{C}$-vector space and so $df_z$ is surjective and $f$ is thus a submersion at $z$.