= Solution
Using the exponential definition of the <hyperbolic cosine> and the <complex exponential function>,
$$
\cosh(x+iy)=\cosh x\cos y+i\sinh x\sin y.
$$
Thus
$$
\operatorname{Im}(\cosh z)=\sinh x\,\sin y,
\qquad x=-\frac b2,\qquad y=\pm\frac{\sqrt{4-b^2}}2.
$$
For real $x$, the <hyperbolic sine> vanishes only at $x=0$. Also $|y|\le1<\pi$, so the only zero of <sine> available to $y$ is $y=0$. The product therefore vanishes exactly when $b=0$ or $4-b^2=0$:
$$
\boxed{b\in\{-2,0,2\}.}
$$
\b[There are three possible parameter values.] At $b=0$ the roots are $\pm i$ and $\cosh(\pm i)=\cos1$ is real; at $b=\pm2$ the roots themselves are real.
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