I would like to create a complex number `c` so that its norm is equal to some number `a` (for the purpose of this question let's assume `a = 1`) if I already have either its real or imaginary part. I know that:
$$\lVert \mathbf{c} \rVert = \sqrt{\sum\_{i=1} ^{n} c\_i \overline{c\_i}}.$$
In this case:
$$\lVert \mathbf{c} \rVert = \sqrt{(a+bi)(a-bi)} = \sqrt{a^2+abi-abi-b^2i^2} = \sqrt{a^2-b^2\cdot{-1}} = \sqrt{a^2+b^2}.$$
As I said, we already know either `a` or `b`, let's say we know that `a = 2`. How can I find what must `b` be equal to if the norm must be `1`? Some simple equations:
$$ \sqrt{2^2+b^2} = 1 \iff 2^2+b^2 = 1 \iff 4+b^2 = 1 \iff b^2 = 1 - 4 \iff b^2 = -3 \iff b = \pm\sqrt{-3} \iff $$

$$ \iff b = \pm\sqrt{3} \* \sqrt{-1} \iff b = \pm\sqrt{3}i $$
Okay, so based on the above eqations, the imaginary part can be $\pm\sqrt{3}i$. So let's say $c=2+\sqrt{3}i$. But clearly my equations are wrong, because:
$$\sqrt{(2+\sqrt{3}i)(2-\sqrt{3}i)} = \sqrt{7},$$ which does not equal to `1`. Please, help me with my confusion and lack of knowledge.