Does there exists any important complex mathematical constant ?
It is known that there exist a large number of mathematical constants that are real numbers, like:
$$\pi, e, \phi, ...$$
Here is the link for much more of them:
https://en.wikipedia.org/wiki/Mathematical_constant
I have never seen and could not find any constant written as: $$c = \alpha + \beta i$$ $$i = \sqrt{-1}$$ $$\alpha, \beta \in \Bbb{R}$$ Why ?
What I mean is that $\alpha$ and $\beta$ can't be also a combination of any other well known constants, so $\alpha$ and $\beta$ should be in some way unique like $\pi$ is unique for themselves and is strongly connected with circle, or like Feigenbaum constants that are strongly connected with bidfurcation and can't be rewriten as combination of any other constatns.
You could argue the most important number in complex analysis is $2\pi i$, because of its appearance in the residue theorem.
Complex roots of unity, i.e. $\{e^{2\pi iq}|q\in\Bbb Q\}$, are also very important, partly because of their role in the theory of polynomials, partly because of complex analysis again. In particular, you often see results of the form $\oint_Cf(z)dz=(1-e^{2\pi is})\int_0^\infty f(x)dx$, which allow us to calculate some integrals of the form $\int_0^\infty f(x)dx$ with a keyhole contour. (If the upper limit is instead finite, doghole contours may encounter the same phenomenon.) You may wish to argue, then, that $1-e^{2\pi is}$ or its reciprocal is therein of more fundamental importance, for suitable $s$, than is $e^{2\pi is}$.