Vector norm such that $\| (1,i) \| = 6$ 
Is there a norm $\| \cdot \|: \mathbb{C}^2 \rightarrow \mathbb{R}^+$ such that $\bigg\| \pmatrix{1\\i} \bigg\| = 6$?

I have come across this question and I don't really know what to look for here. Trying out random norms to see if one of them give $6$ doesn't look like a good idea, and I can't think of any properties of norms that the function in the example would not obey to prove such norm doesn't exist.

Consider
$$\Vert x \Vert = 3\sqrt{x^H x}$$
where $x^H$ is the transpose - conjugation operation, i.e. Hermitian operator. This norm satisfies the three basic norm properties, i.e. 


*

*$ \Vert x \Vert = 0 $ only if $x  = 0$

*For $\alpha \in \mathbb{C}$,$ \Vert \alpha x \Vert = 3\sqrt{(\alpha x)^H \alpha x} = 3 \sqrt{ \vert \alpha \vert^2 x^H x } = \vert \alpha \vert 3\sqrt{  x^H x } = \vert \alpha \vert \Vert x \Vert$

*$ \Vert x + y \Vert \leq \Vert x  \Vert + \Vert  y \Vert $ 

