This thread finally inspired me to learn GLSL. I tried to implement the multi-pass Mandelbrot algorithm described here, where we render to alternating buffers, with the other buffer holding the values from the previous iteration.

uniform sampler2D input;
uniform float curIteration;
void main ()
{
  // Lookup value from last iteration
  vec4 inputValue = texture2D(input, gl_TexCoord[0].xy);
  vec2 z = inputValue.xy;
  vec2 c = gl_TexCoord[0].xy;
...  
}

The value of c at this point is gotten from the texture coordinate, but then the same texture coordinate is also used to index into the texture to look up the value from the previous iteration. How does that work?

I think you're thinking of the single pass algorithm, where you just do a loop for every point in the complex plane.

In the multi-pass algorithm I linked to, you initialise a buffer with the initial z values. Then you compute a single iteration into another buffer: z := z^2 + c. In the code above, the value of c is given directly by the texture coordinate, so how can the same texture coordinates also lookup the value of z in the other buffer?
Either I'm missing something, or that code is wrong.

EDIT: never mind, I found the correct code here. The article is wrong.

Yep, I understand that, just not the specific code in the article. Can you clear this up for me?

  vec4 inputValue = texture2D(input, gl_TexCoord[0].xy);

  vec2 c = gl_TexCoord[0].xy;

If I understand correctly, the first line implies that you should provide texture coordinates that range from (0,0) to (1,1). On the second line there is no scaling and offsetting, so c is restricted to 0+0i to 1+1i.