  We prove that on every compact Riemann surface $M$ there is a Cantor set $C
\subset M$ such that $M \setminus C$ admits a proper conformal constant mean
curvature one ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space
$\mathbb{H}^3$. Moreover, we obtain that every bordered Riemann surface admits
an almost proper $\mathrm{CMC\text{-}1}$ face into de Sitter $3$-space
$\mathbb{S}_1^3$, and we show that on every compact Riemann surface $M$ there
is a Cantor set $C \subset M$ such that $M \setminus C$ admits an almost proper
$\mathrm{CMC\text{-}1}$ face into $\mathbb{S}_1^3$. These results follow from
different uniform approximation theorems for holomorphic null curves in
$\mathbb{C}^2 \times \mathbb{C}^*$ that we also establish in this paper.
