Since $d^2=|a-b|^2=|a|^2+|b|^2-\bar a b -a \bar b$ and $d$ is the positive distance between the vectors $a$ and $b$ it is natural to interpret the angle $\theta$ defined by $\bar a b + a \bar b = 2|a||b|cos(\theta)$ as the angle between the vectors $a$ and $b$.

Since $d^2=|a-b|^2=|a|^2+|b|^2-\bar a b -a \bar b$ and $d$ is the positive distance between the vectors $a$ and $b$ it is natural to interpret the angle $\theta$ defined by $\bar a b + a \bar b = 2|a||b|cos(\theta)$ as the angle between the vectors $a$ and $b$.

Per @BySymmetry's suggestion, note too that $\bar a b + a \bar b = 2 \Re[\bar a b]$ and, since $|\Re[z]| \le |z|$ for any complex $z$, we also have $\frac{|\bar a b + a \bar b |}{2 |a||b|}\le 1$ and we can define a true (real) angle $\theta$ between them by $cos(\theta) = \frac{\Re[\bar a b]}{|a||b|}$.