Let us consider the existence of a zero vector, as required by Property Z of a vector space. The “regular” zero vector fails: $\colvector{x\\y}+\colvector{0\\0} = \colvector{y\\x}\ne\colvector{x\\y}$ (remember that the property must hold for every vector, not just for some). Is there another vector that fills the role of the zero vector? Suppose that $\zerovector=\colvector{z_1\\z_2}$. Then for any vector $\colvector{x\\y}$, we have
\begin{align*}
\colvector{x\\y}+\colvector{z_1 \\z_2}
&= \colvector{y + z_2\\x + z_1}
= \colvector{x\\y}
\end{align*}
so that $x = y + z_2$ and $y = x + z_1$. This means that $z_1 = y - x$ and $z_2 = x - y$. However, since $x$ and $y$ can be any complex numbers, there are no fixed complex numbers $z_1$ and $z_2$ that satisfy these equations. Thus, there is no zero vector, Property Z is not valid, and the set $\complex{2}$ with the proposed operations is not a vector space.