Representation of lie algebra of SL(2,C)

Lie algebra $$\mathfrak{sl}(2,\mathbb{C})$$ consists of all 2x2 complex
traceless matricies. The space of these matricies is 6-dimensional vector space
over real numbers field but is 3-dimensional space over complex numbers field.
Number of different representations of this algebra depend on how we look at
this algebra. If we assume that it's over complex number then it's just complexification
of $$\mathfrak{su}(2)$$ (all representation might be indexed by one integer or
halfinteger number). However if we treat it as space over real numbers then its
representation are index by pair of (half)integer numbers.