The $d$-mandelbrot set is defined as the set of $c$ such that the iterations of
$$z \mapsto z^d + c$$
starting with $z=0$ is bounded in absolute value. Here is a picture of the mandelbrot sets from $d=0$ to $5$:
My question is, if you make this a fractal in the space $\mathbb C \times \mathbb R$, where the first parameter is $c$ and the second is $d$, is it still a fractal in the $d$ dimension? (I guess first of all, how do you define fractal-ness with respect to a certain dimension.)
The real number $d$ is not the dimension of this fractal. It only indexes the generalizations of the Mandelbrot fractal, whose boundary and itself has both two dimensions, like those generalizations. The bulk space is 2-dimensional, because of this, the immersed fractal set cannot have more than two dimensions.