s_{i} = a,-a,…; a = -0.02i

Center: 0; Zoom = 0.4

Recap: The quartic polynomials affine-conjugate equivalence classes can be parameterized in many different, yet reasonable, ways. All of these are mathematically equivalent. However when drawing fractals, they generate different pictures. (It is not surprising, mathematicians think a torus and a coffee cup are the same thing.)

How big is the quartic parameter space?

Quadratics have a single complex parameter. It is two dimensional. We can see the whole parameter space on a plane. Quartics have three complex parameters, and so live in a six dimensional space. We cannot visualize it.

In this six dimensional space resides a six dimensional solid called the “connectedness locus”. (The Mandelbrot set is the 2 dimensional case.) We can cut out two dimensional slices and make quartic fractals. Typically the slices would be made perpendicular to the axis of one of the parameters. You might get fancy and tilt the slice at up three different angles (one for each parameter). But in all cases we only see very small part of the parameter space.

All of the different parameterization and all of the different slices will generate interesting pictures. But in this vast space of parameterization formulas and slices, you will never see the two-step sequence fractals.

You would either need to create a twisted curving non-linear slice, or specifically design the parameterization formula.

One small technicality, the connectedness locus consists of the points where the Julia set is connected, or eqivalently where the iteration of **all** critical points are bounded.
Most of the pictures here examine the behavior of a single critical point.