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The OP was looking for the 1d equivalent rather than an arbitrary slice, and since complex numbers just extend real numbers, R^2 + C seems to be the rather obvious 1d analog.
-Mandelbrot is connected, the line segment -2 to 0.25 is connected
-Mandelbrot inside has dimension 2, this line has dimension 1
The Mandelbrot has border points at the connection with the first great circle, and more further to the head in ever increasing numbers. So R^2 + C  has an increasing number of border points (as R goes towards -2), in fact I would conjecture that it has so many border points that the border has fractal dimension 1. So in fact the line is rather a set of lines with border points (but no space) between them. So..
-Mandelbrot border has dimension 2, this line border has dimension 1

It is actually a pretty interesting shape, and I'm in no doubt that this is the 1d equivalent.
The lengths of the connected line segments don't follow a simple pattern, as can be seen by plotting the bifurcations horizontally on the logistic map.

Well you can if you like, and it might be more interesting. I'm just commenting on what the direct 1d analog is.
Besides, this line segment is interesting. Imagine a line going diagonally along a chess board like a bishop on black squares. At the exact point it crosses to the next square, is the colour of that point black or white? It can be neither. It is fair to define this diagonal line as single segments of length root 2 attached by border points to each other. Each being the single 0d point where the horizontal borders cross.
So R^2 + C looks like this -------+----+--+------+----+--+-+--+-+  The line segments are connected in the same way that the black squares on the chessboard are connected and the same way that the mandelbrot insides are connected. But the imaginary axis of the mandelbrot is not connected, so is not an equivalent fractal in my opinion.

The R^2 + C also shows the decimals of PI when calculating the number of iterations to escape as shown in this table:
"here's a table for points of the form (.25 + X, 0)

X               # of iterations
1.0               2
0.1                8
0.01              30
0.001               97
0.0001       312
0.00001       991
0.000001       3140
0.0000001       9933
0.00000001       31414
0.000000001   99344
0.0000000001 314157"
This is just like the 2d mandelbrot does.

What seems more interesting than the similarities are the differences. The border of the 2d mandelbrot is completely non-smooth fractal, the _border_ of the 1d mandelbrot isn't connected (so is dust) and, looking to 4 dimensions, the quaternion mandelbrot border is overly smooth.
This is interesting and could be explained by saying that the ^2 operation does not have enough degrees of freedom in 1d, has just the right amount in 2d and it doesn't take up all the degrees of freedom in 4d (hence smoothness). This would explain why the mandelbulb requires two operations (on longitude and latitude) and why a properly fractal 4d mandelbrot would require rotating in 3 directions.

Of course this all kind of undermines my point because it means R^2 + C isn't the true 1d analog... the true equivalent would have a connected border of fractal dimension 1 and the line segments should also be connected. I'm not sure such a shape exists.

I think 1d mandelbrot would look like the 2d mandelbrot at the line i0. And i0 so happens to contain an unbroken line. (If you do not know what I mean by i0, y0.)