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I have changed the structure of RC3 to unify the cycle basis (g) and (h) in a single that would be the conventional real numbers, in such a way that the square of a number z=x₀+x₁i+x₂j which belong to RC3 would be:


and the module of z would be: |z|=sqrt(x₀²+x₁²+x₂²)

with these changes, I made a small program in visualbasic with which I have obtained a series of sections of the Mandelbrot3D set thus generated.
As you can see in the attached images in the plane XY(z=0) gets the conventional Mandelbrot set. As we move away from it, the symmetry with respect to the X axis in the direction of the axis Y disappears, although  there is a symmetry with respect to the XY-plane in the Z direction. I also include an image of the silhouette of the complete set.

The code that I have generated the set is this if it is useful:

Public Class mandel3D
    Dim cx, cy, cz As Double
    Dim zx, zy, zz, x, y, z As Double
    Dim iternum, iter As Short
    Dim bailout, modulo As Single
    Dim xp, yp As Single
    Dim col As New System.Drawing.Pen(System.Drawing.Color.ForestGreen, 1)
    Dim formGraphics As System.Drawing.Graphics

    Private Sub Comenzar_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Comenzar.Click
        bailout = 2
        iternum = 1000
       
        For cz = -2.2 To 2.2 Step 0.005
            For cy = -1.2 To 1.2 Step 0.005
                For cx = -2 To 1.2 Step 0.005
                    modulo = 0
                    zx = 0
                    zy = 0
                    zz = 0
                    iter = 0

                    While iter < iternum And modulo < bailout
                        x = zx * zx - zy * zy + cx
                        y = 2 * zy * zx + zz * zz + cy
                        z = 2 * zz * zx + 2 * zy * zz + cz
                        zx = x
                        zy = y
                        zz = z
                        modulo = Math.Sqrt(zx * zx + zy * zy + zz * zz)
                        iter = iter + 1
                    End While

                    If modulo < bailout Then plotset()

                Next cx
            Next cy
        Next cz

    End Sub


    Private Sub plotset()
        Dim centrox, centroy As Integer

        centrox = 200
        centroy = 200
           
        xp = centrox - (0.8660254038 * cx - 0.7071067812 * cy) * 150
        yp = centroy - (cz - 0.5 * cx - 0.7071067812 * cy) * 150
        formGraphics.DrawLine(col, xp, yp, xp + 1, yp + 1)
    End Sub

    Private Sub mandel3D_Load(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Load
        formGraphics = Panelplot.CreateGraphics()
    End Sub
End Class

From this new perspective I will also explore RC4 and represent only the complex part, now I'll tell.

Greetings Paco Fdez.

Coincidence. I tried this too. But using Chaos pro. It have pretty usefull compiler, some 3D engine, and I alredy have alike formula, so just needed to switch my equations with posted. Could be that there are some mistake in code or in formula as 3D looks at the time producing standart 2D mandelbrot. In 2D Chaos pro just cuts off unused quaternion parts, so write whatever in j and part, and it still will generate mandelbrot. So rewriten code to loop all variables and not just quaternionic z.

Bailout conditions included too.

In 2D formula 1 generates mixed inside and outside in real positive side. The colouring aren't the most beautifull, but it nicely accents features. Both formulas produced similar results, what means, that at least there is no accidental mistakes. Much the same as above. All including julia versions have the same streching.

I meant, way how cyclic numbers are implemented could contain mistake so the streching along y=-x. Maybe conditionals if, then, else, or some other non ordinary aproach...

Don't want to steal the thread or distract from seeking what everyone here are seeking. Here I optimised and tuned formula error I made. Negative equation influences just bailout, and not final z.
Allways the squares and crosses. Post function abs create some teuton orden fashioned burning ship. Features revealed by Kali suggested method using low iterations and exp smoothing.


Maybe the same thing deffined with higher powers would generate 5, 6, 7 sided figures, just that equations of it definetely aren't obvious. It looks that critical point is some x=0.7, which generates the most mandelbrot like thing.


Here dgoes Chaos pro:
Today expires my UF, but here is UF code, too.

In keeping with the simplification that I already commented of the ciclo-bases (g) and (h), I have represented the RC3 and set some RC4 to some fixed value x and drawing only the complex part. I include images and code in basic with which they have generated.


As you can see my ability as a programmer it is rather limited.

Best regards

Result looks somewhat stretched and irregular, indicating that it may have some problems..

Maybe using unused 4D (all 5 number parts) could generate more nice mandelbrot set. At least hypothesis seems to be pretty logical and completely fit with exponent function colouring observation.