  Let $q=|q|e^{i\pi\theta},\,\theta\in(-1,1],$ be a nonzero complex number such
that $|q|\neq 1$ and consider the compact quantum group $U_q(2)$. For
$\theta\notin\mathbb{Q}\setminus\{0,1\}$, we obtain the $K$-theory of the
$C^*$-algebra $C(U_q(2))$. Then, we produce a spectral triple on $U_q(2)$ which
is equivariant under its own comultiplication action. The spectral triple
obtained here is even, $4^+$-summable, non-degenerate, and the Dirac operator
acts on two copies of the $L^2$-space of $U_q(2)$. The Chern character of the
associated Fredholm module is shown to be nontrivial. At the end, we compute
the spectral dimension of $U_q(2)$.
