How can we check that for a given norm, we can found an inner product?

Question

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers.
Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner product? Justify.

Is it enough to show that it satisfies the parallelogram law?

Please explain in detail. Thank you.