  We construct two different families of properly Alexandrov-immersed surfaces
in $\mathbb{H}^2\times \mathbb{R}$ with constant mean curvature $0<H\leq \frac
1 2$, genus one and $k\geq2$ ends ($k=2$ only for one of these families). These
ends are asymptotic to vertical $H$-cylinders for $0<H<\frac 1 2$. This shows
that there is not a Schoen-type theorem for immersed surfaces with positive
constant mean curvature in $\mathbb{H}^2\times\mathbb{R}$. These surfaces are
obtained by means of a conjugate construction.
