  We obtain a Runge approximation theorem for holomorphic Legendrian curves and
immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open
and compact Riemann surfaces, and we prove that the space of Legendrian
immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected.
We also show that holomorphic Legendrian immersions from Riemann surfaces of
finite genus and at most countably many ends, none of which are point ends,
satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem,
we infer that every open Riemann surface embeds into $\mathbb{CP}^3$ as a
complete holomorphic Legendrian curve. Under the twistor projection
$\pi:\mathbb{CP}^3\to \mathbb S^4$ onto the $4$-sphere, immersed holomorphic
Legendrian curves $M\to \mathbb{CP}^3$ are in bijective correspondence with
superminimal immersions $M\to\mathbb S^4$ of positive spin according to a
result of Bryant. This gives as corollaries the corresponding results on
superminimal surfaces in $\mathbb S^4$. In particular, superminimal immersions
into $\mathbb S^4$ satisfy the Runge approximation theorem and the Calabi-Yau
property.
