This are different things. The Mandelbrot set iterates z <- z² + c usually with z initialized to zero and c varying across the complex plane with each value corresponding to a pixel in the image. But you can also vary the initial value of z and then the standard Mandelbrot set is a two dimensional slice through a four dimensional generalized Mandelbrot set. You can add another degree of freedom by varying the exponent. The Hausdorff dimension on the other hand characterizes the fractal structure itself, not the number of degrees of freedom the fractal is defined over.
