The biggest speed up will come algorithmically...

If you remember fractint on the PC, it was already very fast and then they added the line following mode, where it decides which pixels to calculate by following a line until the number of iterations changes, then does all the points in that area, and then tries to follow the edge of the region with the same iteration count. When it's made a completely enclosed loop, it can flood fill the area.

There are some gotchas - it's possible to miss some fractal features if the box you trace completely contains it, which is most obviously true for a zoomed-out mandelbrot. Also, when you're zoomed in a lot, there will be areas where this entirely breaks down - in the interesting areas, the iterations per pixel will be changing a lot, and you'll need to give up edge tracing in these parts.

As an aside, you might want to also look at doing Julia sets. These are also done using the equation z'=z*z+c, but instead of c being the initial coordinate, c is a constant across the entire set. You can actually think of the Mandelbrot set as a specialism of the Julia set that's useful for finding interesting areas, but if you find an interesting area of the Mandelbrot set, the Julia sets around there will also be very interesting and you can get some really pretty animations by calculating all the Julia sets with c going from one interesting point on the Mandelbrot to another.