Biharmonic function with a constant modulus

Question

A bi-harmonic function $u:U\to C$, where $U$ is an open subset of the complex plane $C$ is a solution of the equation $\Delta^2u=0$. Can a nonconstant bi-harmonic mapping have a constant modulus in an open set?

Example: The mapping $f(x)=x/|x|$ is a bi-harmonic mapping of the space $R^3$, so the answer to the above question for the space is YES.

Answer 1

It seems the answer is Yes as can be seen by the example $f(z)=z/\bar z$.