Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.
How is this possible with one vector? Would we just find 3 orthogonal vectors to this one and then divide each by the norm of itself? Can we use the Gram-Schmidt process?