I feel that I need to mention Quaternion Complex numbers, because they are seen as the next step in complex maths in dimensions greater than two. I was unsatisfied with the models they produced when I used them to look for the 3D Mandelbrot Set. Using Quaternion maths gave me a 3D set but it was as if it had been turned on a lathe, with all cross sectional planes though the diameter the same, the Mandelbrot set. I was unsatisfied because it was to regular, and I was expecting bubbling cloud-like shape, irregularly complex in all dimensions of viewing.

#### Different Complex Numbers

With the understanding that [imaginary number i] can be represented by rotation of 90° about the origin. I defined [imaginary number j] as a rotation of 90° about the origin and a rotation of 90° with respect to [i].

So with this in mind I picked up a cube, to help me visualise the rotations, and then recorded the results.

I rotated it upwards 90° [i], then following it by turning it a different 90° backwards [j], I defined the result:

i × j = -j

Then starting again

I rotated 90° backwards [j], followed by 90° upwards [i].
This gave me the result:

j × i = i

It is the use of these results when applied into the [Mandelbrot algorithm], that produce the [Bristor set]. Imaginary numbers in additional dimensions can be derived by substituting in the new number in the above equations.

#### The Bristor Set or the Generalized Mandelbrot Set

I have always found it a bit strange and a bit egocentric to call this set after myself, but I feel this is one of my main life changing acheivements. The set is described by the [M-Set algorithm] with the application of my derived numbers. It is though the sets poetry that for me justifies this branch of complex mathematics for me.

The title 'the Generalized Mandelbrot Set' is given to M-Set drawn with [Quaternion] numbers.