= Solution
Writing $z=x+iy$ and comparing <real parts> and <imaginary parts> gives $\sin x\cosh y=\cosh2$ and $\cos x\sinh y=0$. A real solution $y=0$ is impossible since $\cosh2>1$. Thus $\cos x=0$, and positivity forces $\sin x=1$. Consequently $x=\pi/2+2k\pi$ and $\cosh y=\cosh2$, giving
$$
\boxed{z=\frac\pi2+2k\pi\pm2i,\qquad k\in\mathbb Z.}
$$
The right <Argand diagram> in the preceding figure shows representative points on the lines $\operatorname{Im}z=\pm2$; each row continues with spacing $2\pi$.
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