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I make a very simple change to the Mandelbrot set.
instead of z->z^2+c
I do: z -> z^2*(z-a)^3+c.
"a" is a constant that does not vary in space at all (it is set by the user). I used a = 1 (other values of a act similarly, unless a is precisely zero).

Many regions look like perturbation:
(http://www.developmentserver.com/coim/perturb.png)

But then (unlike perturbation) intact Mandelbrot sets, both z^2 and z^3, appear all over the place. These sets seem to appear no matter where we zoom (as long as we can see the boundary of course):
(http://www.developmentserver.com/coim/save.png)
note lack of connectivity in the decorations:
(http://www.developmentserver.com/coim/save2.png)

Like the normal Mandelbrot set, any flyby's come back to "haunt" you every time you look for a baby set.
Things get really interesting when you combine features from different powers. In this case a close flyby to the end of the needle on a miny z^2 m-set is repeated in 3 fold symmetry upon diving toward a z^3 mset:
(http://www.developmentserver.com/coim/3fold.png)
Deeper in, the needle motif becomes the background of the z^3 mset in the center:
(http://www.developmentserver.com/coim/3fold2.png)
This 3-fold needle pattern would never be seen in either the standard z^2 or z^3 msets. More complicated polynomials are possible, and the multiplicity of the root(s) will determine the power of the mset(s) contained inside.

If you want to explore this, you can use orbit traps to easily locate the somewhat rare z^2 msets.
Ultrafractal 5 formula:
http://www.developmentserver.com/coim/formula.html (http://www.developmentserver.com/coim/formula.html)

Welcome, kjknohw! Your post is interesting. You might want to peruse some related work (http://www.fractalforums.com/new-theories-and-research/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/) by fracmonk.

Thanks. They Managed to get coexistence with simply connected fractals  :o!