So how many dimensions does an electron have? It depends.

The electron is described by the Dirac equation. I don't understand the notation at all, but all I cared about was the space in which the equation is valid. The answer is: four complex dimensions.

So you could say it was eight dimensional, but my guess is that it isn't really. Quantum objects are related to waves. When used to describe waves, complex variables are usually of length one. That takes away one degree of freedom, leaving only one degree. Essentially it is polar notation with a constant radius, so the only variable is the angle around a circle. Or perhaps better you could say that the only variable is the phase of the wave.

Electrons are elementary objects so the only possibility is isoclinic rotations. This reduces the degrees of freedom wonderfully. All electrons have the same basic wavelength, so the only variables remaining are the phase and spin. Spin has exactly two possible values. These two correspond to the two chiralities of an even-dimensional rotation. (I don't think it is actually rotating. IMO nobody knows what it is actually going on.)