No nontrivial estimates, at least if the definitions are right.

Say let's define
$$\ell_\alpha c=\sup\,\left\{\,\sum_{t_0<\dots<t_n}|c(t_i)-c(t_{i-1})|^\alpha\,\right\}$$
and
$$\dim c=\sup\{\,\alpha\mid \ell_\alpha c<\infty\,\}.$$

Then from Pythagorean theorem we arrive to
$$\dim c=\max\{\dim x, \dim y\}.$$