It is clear from the symplectic group isomorphism $$SL(2,\mathbb{C}) \cong Sp(2, \mathbb{C}) $$ that there is an $SL(2,\mathbb{C})$ invariant symplectic form on $\mathbb{C}^2$.
My question is whether or not this invariant form is enough to show the existence of an anti-linear $SL_2$ equivariant map, as it is in the compact case.
Another way to ask this, is whether or not the two irreducible 2d representations of the group are self-conjugate reps? I know for non-compact representations duality and conjugacy properties bifurcate.
EDIT: The answer to this question is that the existance of an invariant form and the existance of an invariant anti-linear map, are only "the same fact" when the relevant group is compact.
Note importantly that reps being conjugate to each other and being quaternionic are features of real Lie algebras and their representations. However, $\mathfrak{sl}(2,\mathbb{C})$ is famously isomorphic to $\mathfrak{so}(3,1)$ as a real Lie algebra so we can look at this in two ways.
As a complex Lie algebra $\mathfrak{sl}(2,\mathbb{C})$ has a single complex 2-dimensional rep which is naturally self-dual as the presence of an invariant bilinear form forces. There is no sense here in which we can talk about this rep being self-conjugate or quaternionic.
Meanwhile as a real Lie algebra $\mathfrak{so}(3,1)$ has two complex 2-dimensional reps (the half-spin reps). They are each self-dual but they are conjugate to each other. As mentioned on Wikipedia, there is a Hermitian structure over the pair of them together (but not individually).