3D complex plane analog?

[mrmidget81]

 I'm brand new to these forums, so this has probably already been thought of and posted, but just in case...

From what i gather, if there were some 3D version of the complex plane, a madelbulb would be very easy to create.  I thought of this method, but i'm nowhere near a good enough programmer to explore this, so i'll leave it up to you guys

in a complex plane, if you start exactly on the real number axis, as you iterate through the mandelbrot function, you stay on that same axis.  But, if you start on the axis with just imaginary numbers, when you iterate through the function you jump away from the axis and into the plane.  This to me is the essence of the complex plane - that the two axes interact, but are not exchangeable.

In the 2D complex plane, you can represent any point with a+bi. So, let's imagine that in a 3D space, we can represent any point's location with a+bi+c$.  (i use $ for simplicity, no particular reason).  So, if we imagine the behavior of the $ units relative to units of i as being similar to the behavior of i relative to real numbers, we can say that, when you square c$, you get (c^2)i.  In this way, the $ axis interacts with the i axis (and by extension the real number axis) even though the i and real number axis cannot interact with it.

I'd be interested to see how this translates into a visual form.  One of the interesting things about this is, along the original complex plane the new $ axis has no effect, so when $ is at 0, the shape of the mandelbrot is unchanged.  If nothing else, this could give a pretty cool visual.

a few variations:
have $ squared go straight to the real number plane instead of through i (although i think that would probably just look like a rotated mandelbrot around the real number axis)
add a fourth dimension (one being represented by time obviously for visualization), where the $ and the fourth dimension act like a complex plane, and don't interact with the original complex plane.
give some symbol to the real number plane (% let's say) so when you square any value with %, it gives you $ (this wouldn't really follow the one-way interaction thing i described, but who cares as long as it looks cool) (also, someone could try this on a regualr 2D complex plane to see if it might look cool)

[Timeroot]

There have been many discussions, I think, of different possible number systems. The question that may or may not make your original is: what is $*i? What about $*1? Many people have made tables like the following:

   |  1  |  i  |  $
1 |  1  |  i  |  $
i  |  i   | -1 | 
$ |  $  | | -i

Depending on the spaces, you can get totally different results. More commonly people will use j and k, instead of $ and %. Some people have number systems where (a+bi+cj)*(x+yi+zj) doesn't equal (x+yi+zj)*(a+bi+cj). For example, i*j=-i and j*i=-j. You might want to go onto wikipedia and read about "Quaternions" and "Hypercomplex Numbers" if you're interested in this. 

And one last thing: WELCOME! 

[mrmidget81]

thanks a bunch!  i didn't even think about that little problem (or i guess big problem).  I'll definitely read up on those though.