To sum up, you can't ever see a mathematically accurate picture of a whole fractal, that doesn't just require infinite accuracy in calculation but also infinite resolution in display - similar to not seeing the micro or nano structure of say a leaf whilst looking at it at macro size, or at the whole plant.
However one can always render a view that is truly accurate *to a given resolution* i.e. at that resolution the view you produce is pixel accurate to the real thing (or at least as close as makes no visual difference), to do this well usually means up to at least 3*3 oversampling especially in the denser areas of detail.
Of course there's still the calculation accuracy so for a given view and a given calculation method the maximum resolution that will produce a visually accurate representation of the "true" fractal has an upper limit (essentially such that the calculations must cover up to the accuracy required by the oversample resolution).
With respect to 3D vs. 2D one can never get as high a resolution accurately for a given fractal simply because of the greater inherent errors due to the extra calculations involved i.e. the oversample limit resolution is more restricted, at the moment to a point where rendering 3D views using the GPU (or indeed float or double generally without sophisticated arbitrary accuracy) doesn't get you very high resolutions.

Just to add that with respect to the deepest zooms produced on 2D fractals I'm pretty sure that if the whole Mandy were the size of the known universe then a single pixel's width would be magnitudes less than Planck's length i.e. to see the whole at that scale requires a screen bigger than the whole universe with a pixel size such that even photons are considerably wider than a pixel !

no need for deepest zooms zo achieve this. the scale range from size of universe to planck size is 10e61, so any zoom going deeper than 10e61 (which today is pretty unimpressive) has a pixel size smaller than a planck lenght if blown up to the full universe.