What would it look like?
Let's say we have a large extents viewing window and a small cell size, so we have a large and fixed resolution of 'pixels' within the view window. Now we count the occupied cells as we zoom in.
We know from this previous post that for positive dimension sets the count will grow and for negative sets it will shrink. So for an imaginary dimension set we should expect it to neither shrink nor grow, but oscillate. For a complex dimension we expect it to grow according to the real part of the dimension, and oscillate at a rate according to the imaginary part.
(a) counting cells for a line as we halve cell size: 1, 2, 4
(b) same growth as above. It doesn't have to be a curve to have dimension 1
(c) this shape has different growth: 1, 4, 4, 16, 16, 64, 64,.. so has an oscillation
Above (c) shows a shape that is on average one dimensional, but oscillates relative to the growth.
To extract a sinusoidal growth pattern you cannot just look at behaviour at a scale extreme, but instead look at the count across all scales. Firstly we need to make the signed dimension more robust to the inclusion of oscillations:
Secondly we need to extract those oscillations, which give the imaginary dimension:
Acknowledging this necessary multifractal nature, we can give its complex dimension as: