We denote by $SO(2)$ the group of $2 \times 2$ orthogonal matrices of determinant $1$ with real entries. We have a natural representation of $SO(2)$ on $\mathbb{C}^2$ given by matrix multiplication: If $g \in SO(2)$ and $v \in \mathbb{C}^2$, then $g.v$ is given by the usual matrix multiplication, where $v$ is written as $2 \times 1$ column vector.
Now, since $SO(2)$ is abelian, any complex irreducible representation of $SO(2)$ is of dimension 1. Therefore the above representation is not irreducible and hence breaks up into two irreducible components, each of dimension 1. Also, we know that any irreducible representation of $SO(2)$ is classified by an integer $m$, called its highest weight: $\rho_m : SO(2) \longrightarrow \mathbb{C}^{*}$ is given by $R(\theta) \mapsto e^{im\theta}$, where $R(\theta) \in SO(2)$ is the matrix $ \left( \begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{array} \right)$
What are these irreducible components? Also, in general, what is the way in which one should proceed if one hopes to find the irreducible components of a given representation of a Lie group?
Your representation leaves the 2 lines generated by $(1\ \pm i)^T$ invariant.