Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

Question

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in another example, the imaginary part is taken. Why? And how? Please explain it. Thank you.