[EDIT] These groups have been studied for a long time from various viewpoints, so there is a long paper-trail. I'd emphasize however that working over the complex numbers is usually similar to working over an arbitrary infinite field. Finite fields on the other hand occur more often in arithmetic contexts.
Concerning normal subgroups, the problem looks essentially hopeless, just as in the closely parallel situation of $\mathrm{SL}_2(\mathbb{Z})$ (or its quotient the modular group) first studied over a century ago. There the congruence subgroup problem has a strongly negative answer, in a sense eventually made precise by Serre. In the modern study of the congruence subgroup problem (Bass-Milnor-Serre and beyond), the most manageable situation involves higher rank algebraic groups. But Serre did obtain almost definitive results for rings of integers (and closely related rings of arithmetic interest). His paper is now available online through JSTOR: Le probleme des groupes de congruence pour SL2. Ann. of Math. (2) 92 1970 489–527.
While the congruence subgroup problem is studied mainly in the arithmetic setting, the normal subgroup problem is complicated in a similar way because the matrix group (in your case over a polynomial ring in one variable) has a huge number of such subgroups. In the case of the modular group, these arise from the group-theoretic structure as an amalgamated free product of two small cyclic groups. In most of the literature, attention is focused instead on the somewhat better behaved classical (or Chevalley) groups of higher Lie rank than 1.
While the cumulative literature on matrix groups over rings of various types is enormous and spreads out into algebraic $K$-theory, I'll mention a few of the many people involved over the years: E. Abe, W. Klingenberg, A.W. Mason, A.A. Suslin, L.N. Vaserstein, N.A. Vavilov. Here is a randomly chosen paper: A.W. Mason, Anomalous normal subgroups of $\mathrm{SL}_2(K[x])$. Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 143, 345–358. Vavilov has written many papers on general structure theory, in both Russian and English, including some long surveys.