The tetrabrot fractals are 4 dimensional objects, which are generated from the recursion relation:

Tn+1=Tn*Tn+T0

The elements T which do not diverge under this iteration are the members of the Tetrabrot set.  For the case where T is a complex number (2D), this is the famous Mandelbrot set, shown to the right.  This same image is the 1 vs. i or j axes in the hypercomplex tetrabrot, or 1 vs. i,j,or k in the quaternion tetrabrot.
There are a few ways to extend complex numbers to higher dimensions, creating "hypercomplex" numbers.  These numbers can be represented as sums of the "basis unit vectors", which have multiplication tables such as shown below:

Complex numbers:
                1    i
            1|  1   i
             i|   i   -1

Quaternions:
                 1    i    j    k
            1|  1   i    j    k
             i|   i  -1   k  -j
             j|   j   -k  -1   i
            k|  k   j    -i  -1    

Hypercomplex A:
                 1    i    j    k
            1|  1   i    j    k
             i|   i  -1  -k   j
             j|   j   -k  -1   i
            k|  k   j     i   1    
Hypercomplex Tetrabrot
i or j vs k axes:
Hypercomplex Tetrabrot:
1 vs k axes:
Note that the videos below are NOT images of Julia sets, or exponent variations, or other fractal manipulations.  Only T^2+T is used here!  To generate the video, the 2D "slice" of the 4D set is shifted or rotated, in addition to any zoom or color cycling. 
Quaternion Tetrabrot
i or j vs. k axes:
Lukas Saul
lukas.saul[\*at*]unh.edu
Index of /~javaspid/jfract/movies

Tetrabrot 4D Fractal Slice Movies

[VID] IvK2.mpg                19-Jun-2003 01:26  16.7M  
[VID] QtoBrot.mpg             04-Jul-2003 14:53   3.2M  
[VID] axisCycZoomSq2Brot.mpg  19-Jun-2003 00:46  13.3M  
[VID] brotZoom1.mpg           19-Jun-2003 01:48  13.3M  
[VID] brotZoomIvK.mpg         19-Jun-2003 02:02  13.3M  
[VID] cycle1.mpg              19-Jun-2003 00:51   3.4M  
[VID] inJoutK.mpg             19-Jun-2003 01:03   4.5M  
[VID] noZoomSins.mpg          19-Jun-2003 01:35   3.4M  
[VID] noZoom_005shifty.mpg    19-Jun-2003 01:31   5.0M  
[VID] randomshiftbrot.mpg     19-Jun-2003 02:07   4.1M  
[VID] randomwalkbrot.mpg      19-Jun-2003 02:10   3.6M  
[VID] roundabrot.mpg          19-Jun-2003 02:14   3.4M  
[VID] shiftcantorbrot.mpg     19-Jun-2003 02:16   2.4M  
[VID] smallsqtobrot.mpg       19-Jun-2003 02:20   4.5M  
[VID] squaretobrot.mpg        19-Jun-2003 02:32   3.4M
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