Fractals like and are easy to draw because they just involve complex addition and multiplication. What happens if we try to generalize these sorts of fractals to non-integer powers?
We can do this using the identities
The trouble is, we have to pick a value for . Given a parameter we can pick such that (i.e. choose a branch cut of constant argument and a sheet of the Riemann surface): .
Then as we gradually vary the fractal we get will change. For , , increasing will cause to "spiral in" from infinity towards the origin () or out (). When , increasing will have a periodic effect, which will cause the fractal to appear to "rotate" in fractally sort of ways with a period of .
It would be interesting to plot these as 3D images (like these with on the z axis. These would look sort of helical (or conical, for .)
Using poses some additional problems - the attractor isn't bound by any circle of finite radius about the origin so it's more difficult to tell when a point escapes and when it is convergent but just currently very far out. Clearly some further thought is necessary to plot such fractals accurately - perhaps there are other escape conditions which would work for these.