  In this paper the fractional-order Mandelbrot and Julia sets in the sense of
$q$-th Caputo-like discrete fractional differences, for $q\in(0,1)$, are
introduced and several properties are analytically and numerically studied.
Some intriguing properties of the fractional models are revealed. Thus, for
$q\uparrow1$, contrary to expectations, it is not obtained the known shape of
the Mandelbrot of integer order, but for $q\downarrow0$. Also, we conjecture
that for $q\downarrow0$, the fractional-order Mandelbrot set is similar to the
integer-order Mandelbrot set, while for $q\downarrow0$ and $c=0$, one of the
underlying fractional-order Julia sets is similar to the integer-order
Mandelbrot set. In support of our conjecture, several extensive numerical
experiments were done. To draw the Mandelbrot and Julia sets of fractional
order, the numerical integral of the underlying initial values problem of
fractional order is used, while to draw the sets, the escape-time algorithm
adapted for the fractional-order case is used. The algorithm is presented as
pseudocode.
