The Mandelbrot VS the Triceratops fractal in XaoS: what makes the difference?

Many years ago I played with the source code of the XaoS fractal zooming program... I searched for new (fractal) formulae, and viewed how they look like as fractal images. I had the idea that instead of f(z)=z²+c (the iteration formula for the Mandelbrot set), I could use the generalization of triangular numbers (for complex numbers), so I used triangular numbers instead of squares. The formula for triangular numbers is n*(n+1)/2, so the iteration formula for my new fractal has become f(z)=(z²+z)/2+c. A very interesting fractal formed, which I called Triceratops. This was interesting because it was very different from the Mandelbrot set...

The Mandelbrot set

The Triceratops fractal

Of course, we need to zoom in to see the interesting parts of the fractal, like:

These are kinds of images we do not usually see in the Mandelbrot set. Later I realized that this is because the escape condition was too low for the Triceratops fractal... the escape condition for the Mandelbrot set is 4, which means that the iteration stops when |z|²>4, or abs(z)^2>4. In the case of the Mandelbrot set, this is equivalent to saying that the point is not part of the fractal, because it diverges to (some) infinity... and therefore, it is colored by some color different from black (depending on the number of iterations needed to get this result). However, in the case of the Triceratops fractal formula, this condition of |z|²>4 was not enough, so we excluded some points which might otherwise be parts of the fractal (and be black). Increasing that to e.g. 32, we might get a fractal very similar to the Mandelbrot set:

The f(z)=(z²+z)/2+c fractal with escape condition |z|²>32

Zooming in this fractal makes images similar to those of the Mandelbrot set... and conversely, if we reduce the escape condition of the Mandelbrot fractal (to e.g. 1 or 0.8), we may get images similar to those of the Triceratops fractal... Thus the secret of these interesting images lies in the escape condition, not really in the formula.

Zoom in the Mandelbrot fractal but with escape condition |z|²>0.7