Entropy versus influence for complex functions of modulus one

We present an example of a function $f$ from $\{-1,1\}^n$ to the unit sphere in $\mathbb{C}$ with influence bounded by $1$ and entropy of $|\hat f|^2$ larger than $\frac12\log n$. We also present an example of a function $f$ from $\{-1,1\}^n$ to $\mathbb{R}$ with $L_2$ norm $1$, $L_\infty$ norm bounded by $\sqrt{2}$, influence bounded by $1$ and entropy of $\hat f^2$ larger than $\frac12\log n$.