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I use Mandelbulb, so that is where this idea arose.

I am not sure if one or both of these systems has or have already been proposed, but here goes:

Could one take the average number of raysteps for the pixels surrounding a certain pixel and use that average as a preliminary value for the proximity of the fractal surface? If an average of the nearby numbers of raysteps was used, with the calculation jumping up a raystep from the average, then down a raystep from the average, then up two raysteps from the average, then down two raysteps from the average, and so on for infinity could be used until either a surface is reached or the calculation ends. This would, hypothetically, make rendering regions of the fractal that are relatively similar in distance to the viewer render more quickly, right?

This next idea is a bit different. If the average number of raysteps is very high and standard deviation of the number of raysteps for nearby pixels is 0 (as would be the case whenever there is a large area of negative space), the program could just assume that that value will also be negative space. This would require a checkerboard approach to calculation, in which some pixels are calculated first using the method outlined in the previous paragraph and the ones inbetween those pixels are assumed to be negative space  as well if the average number of raysteps is high and standard deviation is 0. This method of "assuming" values could lead to a small loss of quality, I would think, so it may be less desirable than the method outlined in the previous paragraph. This checkerboard approach could also be used for the previous paragraph as well.

Does anyone know if these ideas have any capacity to expedite rendering?

The issue I can see is that with sphere tracing using distance estimates each ray step is a different size, depending on how close the surface is.  So number of steps isn't an indication of position.