Are representations of $\text{SL}(2,\mathbb{C})$ indexed by one half-integer or two?

The Lie group $SL(2,\mathbb{C})$, viewed as a complex Lie group, has irreducible representations of complex dimension $2j+1$ classified by a single half-integer $j\in\frac{1}{2}\mathbb{N}_0$.

The Lie group $SL(2,\mathbb{C})$ is the double-cover of the restricted Lorentz group $G:=SO^+(1,3;\mathbb{R})$. The latter is naturally viewed as a real Lie group in physics.

Its complexification $G_{\mathbb{C}}=SO(1,3;\mathbb{C})$ has double cover $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$, whose irreducible representations are classified by two half-integers (since there are now a product of two $SL(2,\mathbb{C})$ groups). See e.g. this & this Phys.SE posts.

A representation of the complexification $G_{\mathbb{C}}$ is also a representation of the restricted Lorentz group $G$. Conversely, any physically relevant representation of $G$ is expected on physical grounds to be a representation of $G_{\mathbb{C}}$ by analyticity.