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It is surely not a fractal from a strict point of view, but what about the Feigenbaum graph? You can generate it from the real component (bi=0) of the mandelbrot iteration. The graph itself might not be 1-dimensional but it arises from a 1-dimensional parameter. And there are bifurcations all over the place, so there is some sort of self similarity inside too. Maybe not a fractal and not 1-d in the end. But there is definitely a close relation.

I'm too wondering, which one you mean
do you say the Feigenbaum-graph "maybe not a fractal" or did I mmisunderstand something?

The Feigenbaum graph is really a collection of many 1D point sets. Any point on the X axis of the above depiction represents a single parameter, a real number. Above that point, parallel to the Y Axis, the corresponding 1D point set for that parameter is plotted. That point set is the attractor of a dynamic process, i.e. the "infinite limit" of some iterated computation, parameterized with the value from the X position.

Across the whole parameter range (i.e. the relevant interval of the X axis), the corresponding attractors fall into two broad categories. Either an attractor consists of a finite number of isolated points, or the attractor has an infinite number of points that densely fill (part of) the Y axis.

"Simple" attractors with a finite number of points mean that the underlying iterated computation ends up cycling through a fixed list of values. Such an attractor is not fractal.

"Chaotic" attractors with infinitely many densely spread points indicate unstable, "unpredictable" behaviour of the iterated computation. These attractors are usually fractal point sets.

(There is also a theoretical possibility to get an infinite number of non-chaotic points, but I don't know if that case can happen in the Feigenbaum graph. An example of such a point set would be all points at positions 1/n for n any positive integer. But to be relevant to the Feigenbaum graph, an iterated computation would have to visit each point infinitely often - I am not sure if Feigenbaum's formula can do that when iterated.)