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Thanks Syntopia, useful info.
Here you can see the surface with height of -0.25 up to 1:


It really explodes out like a volcano when height increases past 1, so I think I'll call it a volcanic surface fractal.
Around 1.75 the fractal occupies all of space and has a Hausdorff dimension of infinity!

Works - managed to disable that ANGLE!

It is because there is overlap (for height > 0.55), so some points in space have double cover, or triple cover etc. You can do the same thing with the Levy C curve, for 90 degree bend angle it has Hausdorff dimension 2 (log(2) / log(sqrt(2))), for bend angle > 90 degree it is log(2)/log( < sqrt(2) ) so the dimension is > 2.
If you don't allow multi-cover of the space then the max Hausdorff dimension is the dimension of the space.

Here's a higher res of above image:

i get this, but in my eyes it is not really special, because such curves exists ... basically the trivial curve with no movement ... the dot at the center of the c oordinate system would have as well unlimited hausdorff dimension

this brings us to the space filling curves and why it is such a good name but as far as i know the definition of space filling curves is NOT limited to the topografical dimension !?

that iteration looks pretty wild. nice.

There's an interesting critical point, when the height is 0.55209 then all the triangles are isosceles. This is also about the point where the shape is at its most rough without self-intersecting. I'm not sure if those two are exactly the same point but they could well be.