The Mandelbrot set is closely related to a circle, so that's how I will take a stab at this. A circle, mathematically, can be considered a sphere if we define a sphere in a general sense as being the set of all points that are equidistant from the center. In technical terms, a 2D circle is considered an S1 sphere, because it is made out of a 1-dimensional border (line). A 3D ball is considered an S2 sphere because it has a 2d surface. Likewise, the S3 sphere is 4D and so on.

We can't observe 4D objects directly but as 3DickUlus pointed out, we can project down to the next lower dimension. So, for example, you've probably seen the World map where the globe is projected downward onto a circular shape, with the North Pole at the center. The South Pole becomes a singularity that explodes, thus Antarctica surrounds the entire perimeter. Mathematically, this is known as stereographic projection of an S2 sphere onto an S1 sphere. An interesting property of this projection is that even though the continents are distorted, the perfect circles of latitude on the globe are still perfect circles on the map.

It is possible to do the same thing with higher dimensions. Circles on an S3 sphere (4 dimensional) can be stereographically projected down onto an S2 sphere (3 dimensional) where we can see it. The result is a distorted view of the 4D sphere but as before these circles maintain their circular nature, and they are known as "fibers." The collection of these fibers is known as a Hopf fibration. The collection of fiber circles that are mapped to a single line of latitude on the S2 sphere form a torus. So the Hopf fibration can be thought of as nested tori. The circles are known as Villarceau circles

https://en.wikipedia.org/wiki/Villarceau_circles and although in the Hopf fibration there are an infinite number of these circles forming an infinite number of nested tori, you can choose any two circles and they will be interlocked, but no two circles overlap. Here is a great video that demonstrates the relationship between the fibers and the S2 sphere. A single point on the S2 sphere corresponds to a single circle. Notice that the singularity occurs at the north pole, where the Hopf fiber becomes an infinitely big circle and so just looks like a straight line. At the South pole, the torus that is formed collapses down into a circle with a radius of 1. It seems to be a really cool object but back to the original question - I honestly have no idea how to apply this to fractals in a practical way... maybe someone else here can help make that connection.