Performance at first

What is the absolutely fastest way to draw lets say a mandelbrot fractal (fixed number of iterations and other mandelbrot settings) ower whole the screen at current resolution?

store the mandelbrot fractal result as a texture and draw that as a screen sized quad

Painting pixel graphics, is a quad the fastest way? The way which provides the fastest way to the hardware?

If you want to paint pixel graphics with GL it will always be VERY slow.
You should do as zed suggests: render the fractal to a texture and display it on a rectangular mesh(quad).

Use a fragment shader and generate the fractal directly on the graphics card.

With all parameters fixed, building the texture offline and in advance, then loading it as a standard texture and drawing it using standard draw commands will always be the fastest. It’s only going to be 4 vertexes so it won’t matter much what primitive type you use, or whether you use immediate mode, vertex arrays or a VBO.

You won’t be able to zoom/pan/etc on the fractal using this method.

If you need to zoom/etc then you need to build it procedurally in a shader. Once again, the items I mentioned above won’t matter, that’s not where your bottleneck is going to be, and it’s not where any of the heavy work is going to be done. So pick whichever suits your coding style and/or the GL_VERSION you’re targetting.

You can very easily set up parameters that can drag any GPU to it’s knees this way, you you should initially set up for a low number of iterations and ramp it up until you get a good balance between detail and performance.