Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus E$,where $\bar {\Bbb C}:=\Bbb C\cup\{\infty\}$ is the extended complex plane.Denot $A(E)$ the class of all functions that are continuous on $E$ and analytic in the interior of $E$. Let $P_n$, $n\in \Bbb N_0:=\{0,1,2,...\}$, be the class of complex polynomials of degree at most $n$. For $f\in A(E)$ and $n\in\Bbb N_0$, define $$E_n(f,E):=\inf_{p_n\in\Bbb P_n}\|f-p_n\|_E ,$$ where $\|\cdot\|_E$ denote the supremum norm on $E$, and $$P_n(z)=\sum_{i=0}^n{c_i}z^i$$ $c_i \in\Bbb C,z\in\Bbb C,n\in\Bbb N_0.$ By the Mergelyan theorem, we have: $$\lim_{n\to\infty}E_n(f,E)=0\ \ \ \ \ \ (f\in A(E))$$
In the other hand, Euler's formula states that, for any complex number $c$, there is $a_0\in\Bbb R^+$ and $\phi_0\in [0,2\pi]$ for $c=a_0e^{i\phi_0}$.
My question is that,whether there is a function $f_n=A_n(x)e^{iB_n(x)}$, in which $A_n(x)$ and $B_n(x)$ are real polynomials: $$A_n(x)=\sum_{i=0}^n{a_ix^i}$$ $$B_n(x)=\sum_{i=0}^n{b_ix^i}$$ $a_i,b_i\in \Bbb R,x\in\Bbb R,n\in\Bbb N_0$, that is a approximation of a complex polynomial $p_m(z)$？
Namely, does these kind of $f_n$ exist for: $$\lim_{n\to\infty}\inf\|f-f_n\|_E=0\ \ \ \ \ \ (f\in A(E))$$ or for a given $p_m \in P_n$ $$\lim_{n\to\infty}\inf\|p_m-f_n\|_E=0\ \ \ \ \ \ (p_m\in P_n)$$？