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very good question, the answer is: yes! we know, and it does produce interesting results, check out the mandelbulb triplex

yes, the tricomplex numbers and triplex numbers are very different.

I tried the tricomplex numbers a while ago, and did not find anything interesting (just extruded Mandelbrots/Julias - see attached image).

But I only used the multiplication operator - for tricomplex numbers, there is also an unusual modulus operator, which probably should be used instead of the Euclidean length. I tried this, but couldn't get anything but noise with my analytical DE - but perhaps a brute force approach would work here.

Example pic:

I tried using tricomplex numbers with my mandelbrot set and I got the exact same geometry as Syntopia. However, when I paired them with the Julia Set I got nothing. No image at all...

The fact that tricomplex numbers are so boring in comparison to triplex and quaternions, does this tell us anything about the underlying or underpinning mathematics behind these number systems? Slightly off topic, but I was wondering about extending complex numbers to three dimensional space. Is tricomplex numbers it. Is that the way? Or is triplex numbers what we want to be using (Some of the mathematics of triplex numbers look a little "merky" to me and it seems it was created more with the focus of beautiful images instead of mathematical beauty).

What about what William Hamilton said...we need a fourth dimension (k) which we take a slice of (3D). So does that mean true hypercomplex 3D numbers are not possible? If so...why?

And I think the biggest question I have is... If we actually extend complex numbers to 3D space how will we know it is correct. And how will we know what a 3D mandelbrot looks like? What criteria do we use? Shouldn't the way to compute the 3D mandelbrot be synonymous with the 2D way? After all its the same shape. So changing its method of formation would not really be 100% analogous. or... Am I just completely off the mark

Thanks everyone. I am really looking forward to get my questions answered

Farewells,
Julian V.

Interesting. Like Asdam quoted the Wikipedia page on tricomplex numbers was taken down.

I saved the page on desktop. But experimenting with numbers, it looks that extruded property is generated by ixj=1.

If you do (Ar,Bi,Cj)x(Ar,Bi,Cj), in square power ixj is just single part 2*zy*zz. So if 2*zy*zz are in formula for x, very probably that fractal would be extruded.