  This paper brings several contributions to the classical
Forster-Bell-Narasimhan conjecture and the Yang problem concerning the
existence of proper and almost proper (hence complete) injective holomorphic
immersions of open Riemann surfaces in the affine plane $\mathbb C^2$
satisfying interpolation and hitting conditions. We also show that in every
compact Riemann surface there is a Cantor set whose complement admits a proper
holomorphic embedding in $\mathbb C^2$. The focal point is a lemma saying the
following. Given a compact bordered Riemann surface, $M$, a closed discrete
subset $E$ of its interior $\mathring M=M\setminus bM$, a compact subset
$K\subset \mathring M\setminus E$ without holes in $\mathring M$, and a
$\mathscr C^1$ embedding $f:M\hookrightarrow \mathbb C^2$ which is holomorphic
in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic
embedding $F:M\hookrightarrow \mathbb C^2$ which maps $E\cup bM$ out of a given
ball and satisfies some interpolation conditions.
