Weyl spinor representations and the Lorentz group

There are several reasons:

1. The complexified version is semisimple, and hence can be decomposed by the classification of semisimple complex Lie groups. This is much simpler since the field of complex numbers is algebraically closed.

2. From the complexified version one can always go back to the real version by considering in the Lie algebra of generators only the Hermitian elements (invariant under conjugation). Thus nothing is lost.

3. The representation theory of a Lie group and its complexification is closely related. In the present case, this is of great use since the representation theory of the factors of the complexified group is particularly simple, and allows one to deduce the representation theory of the original group.

4. In quantum physics, one wants to work with Hermitian generators rather than antihermitian ones as in mathematics. This already introduces complex coefficients.

5. It is often convenient to look at linear combinations of generators with complex coefficients, which are naturally in the complexified Lie algebra. For example, in the Heisenberg algebra, one gets the creation and annihilation operators in this way.

1. The passage from real to complex is harmless, as the complex linear representations of the complexified Lie algebra are in 1 to 1 correspondence with the real linear representations of the real form. (Given, that we are working with representations on a $\mathbb{C}$ vector space.)
2. As you wrote, the complexified Lorentz algebra can be written as a sum of $\mathfrak{sl}_2(\mathbb{C})$, which has the significant advantage, that the representation theory of $\mathfrak{sl}_2(\mathbb{C})$ is completly understood. See, for instance, this article.
3. Using the representations of these copies of $\mathfrak{sl}_2(\mathbb{C})$, we can label the representations of the complexified Lorentz algebra, and thus those of the Lorentz algebra (see 1.) by pairs $(i,j) \in \mathbb{N}/2 \times \mathbb{N}/2$, which helps when talking about particles 'living in certain representations'. More on these can be found in Knapp - Representation Theory of Semisimple Groups_an overview based on examples.
4. Having classified all the representations of the Lie algebra, you get those of the Lie group, but have to take care of so called 'projective representations'. An overview can be found in this article. Ignoring technical difficulties, it doesn't matter as you would want to allow projective representations when talking about quantum physics anyways. An then, classifying all representations of the Lorentz algebra gives rise to a classification of all possible spin configurations (assuming that spin is described by the Lorentz group). A source for further reading might be D.J. Simms - Lie groups and quantum mechanics, though you should be familiar with basic representation theory if you want to give it a try (as a first read for representation theory, I would recommend B. Hall - Lie Groups, Lie Algebras, and Representations: An Elementary Introduction).
5. The universal cover of the Lorentz group is $SL_2(\mathbb{C})$, viewed as a real Lie group (it is sometimes written as 'Spin$(1,3)$'). The classification of its Lie algebra does indeed classify its representations as it's a simply connected Lie group.