Taken from SubBlue.com
A quaternion Julia set is a four-dimensional equivalent of the standard two-dimensional fractal. By taking a 3D 'slice' through the 4D space it is possible to visualize a solid fractal.
Complex numbers have two components that define a point in a plane and are the core build blocks of fractal calculations. In 1843 the Irish mathematician Sir William Rowan Hamilton developed quaternions as a way of describing complex points in three-dimensional space. The only snag was he had to add a fourth dimension to make the maths work (hence the quad part of the name). Whilst complex numbers are described as the sum of a real and imaginary component: z = a + bi, quaternions are similar but have three imaginary components: z = a + bi + cj + dk
Fractals can be calculated using quaternions with the usual equation zn+1 = zn2 + c where we track which points have a magnitude greater than the bailout threshold. The problem is trying to visualize the set of 4D points in 3D space. At this point I'll direct you to an excellent write-up Keenan has posted that describes the process in detail.