Let $z$ be a complex number of the form $a + bi$ such that $a , b \in \mathbb{R}$. So, we can express $z$ as $$z = | z |(\cos \theta + i\sin \theta)$$
where $|z| = \sqrt{a^2 + b^2}$ and $\theta = \arctan \left(\frac{b}{a} \right)$. Using Euler's formula, we can express $$\cos \theta + i\sin \theta = e^{i \theta}$$
So, substituting the value of $\cos \theta + i\sin \theta$ in first equation, we get $$z = |z|e^{i \theta}$$ $$e = \left (\frac{z}{|z|} \right )^{\frac{1}{i \theta}}$$
So, one can define $e$ as the ratio of $i \theta^{th}$ root of any complex number and its absolute value. Is this definition correct?