Let $f(z) \in \mathbb{C}[z]$ with deg$(f) \geq 1$. Then $\{z \in \mathbb{C} : |f(z)| > 1\}$ is connected.
I know that $f(z)$ is holomorphic, so continuous on $\mathbb{C}.$ I dont think that preimage of connected set under continuous map is always connected.
Right now I just know that $\{z \in \mathbb{C} : |f(z)| > 1\}$ is open since it is exactly the set $|f|^{-1}((1,\infty)).$
Since the given set contains the exterior of a large circle $C$, there is certainly a connected component that contains $\infty$, which also contains the exterior of $C$.
Suppose the statement is not true. Then there will be another connected component $D$, which is inside $C$. Since $\lvert f\rvert$ is continuous $D$ is a compact domain. Apply the maximum modulus principle to $f$ in $D$: $\lvert f\rvert$ takes on its maximum on the boundary of $D$, but $\lvert f\rvert \geq 1$ on the interior of $D$ by the definition of $D$, so $D$ must be constant, which is a contradiction.