= Solution
Write $f=u+iv$ and evaluate all <derivatives> of $\psi$ at $(u,v)$. The real <chain rule> gives
$$
\Delta(\psi\circ f)=\psi_{uu}(u_x^2+u_y^2)+2\psi_{uv}(u_xv_x+u_yv_y)+\psi_{vv}(v_x^2+v_y^2)+\psi_u\Delta u+\psi_v\Delta v.
$$
The <Cauchy-Riemann equations>[Cauchy–Riemann equations] give equal squared lengths $u_x^2+u_y^2=v_x^2+v_y^2=|f'|^2$ and zero cross term $u_xv_x+u_yv_y=0$. Part (ii) removes the last two terms. Therefore
$$
\boxed{\Delta(\psi\circ f)=|f'|^2(\Delta\psi)\circ f=0.}
$$
The composition is $C^2$, so it is <harmonic>. No assumption that $f'$ is nonzero is required.
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