Here is one way.
z = a + b I;
f = (z - 3 I)/(z (z + 1)) // ComplexExpand
Now solve for when the real part is zero:
sol = Reduce[Re[f] == 0, {a, b}, Reals]
Here is a contour plot showing some of the solution:
ContourPlot[Re[f] == 0, {a, -5, 5}, {b, -10, 10},
FrameLabel -> (Style[#, Bold, 18] &) /@ {"a", "b"}, PlotPoints -> 100]
