I can think of two points in the complex plane $z_1=1$ and $z_2=-1$ that are invariant under inversion, which means that
$$\frac{1}{z_i}=z_i$$
Are there any other such points, or is this the whole set? If so, how come these two points are so special to obey this property? Does this mean that the complex plane is in a sense axial-symmetric to the imaginary axis (or real axis)?