 Solution

Write $z=x+iy$ with real $x,y$. The complex hyperbolic sine is $\sinh z=\sinh x\cos y+i\cosh x\sin y$. Its zero real part requires $\sinh x\cos y=0$. If $\cos y=0$, the magnitude of the imaginary part would be $\cosh x\ge 1$, which cannot equal $1/2$. Therefore $x=0$ and $\sin y=1/2$. All solutions areOn the Argand diagram these are two interlaced arithmetic sequences on the imaginary axis; both repeat with imaginary period $2\pi i$.