= Solution
Reflection in the real axis replaces a dipole at $a$ with an image at $\bar a$ of moment $\bar m$. Reflection in the imaginary axis places the image at $-\bar a$ with moment $-\bar m$. Applying both reflections gives the image at $-a$ with moment $-m$. Thus
$$
\boxed{W(z)=\frac{m}{z-a}+\frac{\bar m}{z-\bar a}-\frac{\bar m}{z+\bar a}-\frac{m}{z+a}}.
$$
For example, on the real axis the first two terms are conjugates and the last two are conjugates. On the imaginary axis the first and third terms are conjugates, as are the second and fourth. The potential is real on each wall, so both walls are streamlines. The only singularity in the open first quadrant is the original dipole.
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