Non holomorphic fuction with harmonic real and imaginary parts

Question

I‘m trying to think of a function whose real and imaginary parts are harmonic which isn‘t holomorphic to show that the converse of “a holomorphic function has harmonic real and imaginary parts“ isn’t true but can‘t come up with such a function myself.

Answer 1

$z \to \overline {z}$ is not holomorphic but its real and imaginary parts are harmonic.

Answer 2

If someone said he could not find a counterexample to "Every prime is even" you'd conclude he had literally not tried anything, since what's hard is finding a non-counterexample. Same here:

Assuming "harmonic" means "real-valued harmonic" and we're talking about functions in some connected open set: Say $u$ and $v$ are harmonic. If $u+iv$ happens to be holomorphic let $v_2$ be any harmonic function other than $v+c$ for some constant $c$; then $u+iv_2$ is not holomorphic.