Does any generalization for "famous" math constants (like $\pi$ or $e$) exist.
I know how those constants are useful, but I do not know what property makes them useful.
Is there any definition of those numbers which can include some additional numbers. (Those additional number can be part any super-set of $R$ or $C$).
What property of those constants is making them "special"?
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Perhaps you are looking for non-algebraic ( transcendental ) numbers? Certainly, something shared with pi and e.
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Perhaps you may be interested in ring of periods, recently developed by Kontsevich and Zagier, generalizing the constants you mentioned. More details in the Wikipedia article and its references. According to Kontsevich and Zagier, "all classical constants are periods in the appropriate sense".
Well, we can calculate $\pi$ by $$\pi = \lim_\limits{n\to \infty} \frac{1}{2} * n * \sin{(\frac{n}{360})}$$ And $e$ by using $$e = \sum_{n=0}^\infty (\frac{1}{n!})$$ Those two definitions are somewhat the proof that they can't actually be calculated. However, since they do occure in nature, they are very speciall. Also, the beautiful equation $$e^{i * \pi} + 1 = 0$$ States that they are relative to each other and $\sqrt{-1}$ (an imaginary number), which, once again, makes both of these very special.