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Now that quadrium | flame 1.2 is finally available, time to show off a bit.  These are kleinian limit sets - basically, if you are familiar with how you can make Sierpenski's triangle using the limit set of three affine transforms (where one scales by 50%, the second scales by 50% and offsets x+0.5, and the third scales by 50% and offsets y+0.5), these are the same sort of thing only done with two bilinear transformations* (plus their inverses):

 
http://www.deviantart.com/deviation/59895968/

 
http://www.deviantart.com/deviation/59970373/

 
http://www.deviantart.com/deviation/60004297/

* bilinear transformations are of the form f(x) = (a * x + b) / (c * x + d) where all the values are complex numbers.  Bilinear transforms like this are often called Moebius Transformations or fractional linear transformations, and preserve angles and circles (unlike affine transformations which can distort circles into ellipses).

How do you create these Kleinian Shapes in Ultra Fractal?
I have tried and I have got weird results.

1. I use Ultra Fractal more than any other program out there.
2. I don't have the admin password to my computer to install the software.
Otherwise I would have Apophysis exe
Is there a stand-alone program of Apophysis? If it just runs off the .and not the installer, can use it.
Sorry if  I thought your comment was a little rude. That's how I react to things like that.

If I remember correctly there should be something in the "reb" collections but I can't remember which file or if it's written as a colouring or a main formula.

The latest beta versions don't require installation.
http://sourceforge.net/projects/apophysis/

Only thing is, you will have to create a folder for scripts yourself, if you want to use them.