It might be immodest of me to mention my own work, but there is quite a lot known about the pseudo-holomorphic curves in $S^6$. For example, it is known that the pseudoholomorphic rational curves are all algebraic and have area a multiple of $4\pi$ (and, yes, there are many of them). (It's true that the moduli space is not compact; however, because the area is quantized, the rational curves of a given area, can, in fact, be compactified to yield a compact moduli space, even an algebraic one.) As another example, every compact Riemann surface occurs as a pseudoholomorphic curve (possibly ramified) in $S^6$. (A student of Kevin Corlette, some years ago, proved that you can actually get every Riemann surface as an *unramified* pseudoholomorphic curve in $S^6$.)

For example, see my paper *Submanifolds and special structures on the octonians*, Journal of Differential Geometry **17** (1982), 184–232. (I regret very much the misspelling 'octonians' that permeates that paper.)

**Added references:** (January 2020) Recently, I was asked some questions about pseudo-holomorphic curves in $S^6$, and that inspired me to look at some of the literature that has been generated in the nearly 40 years since I wrote the paper above. A fair amount has been worked out, and anyone who wants to know more about the subject would benefit from consulting a few of the following papers:

J. Bolton, L. Vrancken, and L. Woodward, *On almost complex curves in the nearly Kähler $6$-sphere*, Q. J. Math. Oxford (2) **45** (1994), 407–427.

H. Hashimoto, *Deformations of super-minimal $J$-holomorphic curves of a $6$-dimensional sphere*, Tokyo J. Math. **27** (2004), 285–298.

J. K. Martins, *Superminimal surfaces in the $6$-sphere*, Bull. Braz. Math. Soc. (N.S.) **44** (2013), 25–48.

L. Fernández, *The space of almost complex $2$-spheres in the $6$-sphere*,
Trans. Amer. Math. Soc. **367** (2015), 2437–2458.

M. Dajczer and T. Vlachos, *A representation for pseudoholomorphic surfaces in spheres*, Proc. Amer. Math. Soc. **144** (2016), 3105–3113.

This is by no means a complete list of relevant articles, but these do cover a range of interesting developments of the results in my 1982 JDG paper.