There is more to a beautiful equation than the shape of the symbols used to describe it. The beauty of Euler's identity is the relationship between 5 fundamental constants (0, 1, e, i, pi). It's simple, elegant and far reaching.
The relationship is the same regardless of the notation.
Mostly serious comment: I'm not sure why the form e^{i\pi}=-1 isn't better. Only it doesn't have 0, but is it worse, less beautiful? (It doesn't seem to have the same "relationship between 5 fundamental constants", although it adds the negative number realm, to the imaginary and transcendental–neat.) Would E-mc^2=0 be similarly be better than E=mc^2, because it has an additional "fundamental constant"?
People fetishize the formula but the beautiful idea is that multiplication/exponentiation can be expanded in such a way that it describes oscillatory relations. This is how eg. eigenvalues get to play a role in models of harmonic resonance. Or how AC impedance naturally generalizes DC resistance.
Try to imagine complex interest rates. Now try to make them matrix-valued. It works. It all works.
I don't want to start an argument about tau versus pi, but I like a tau form (or modification) of Eulor's Identy: [e^(ikτ) = 1] for all integer values of [k]. This gets across rather well that this formula expresses a complete turn around a unit circle. You can't get something quite equivalent using pi.
I even more prefer the full form of Euler's Formula: [e^(ix) = cos(x) + isin(x)]. The real beauty of Euler's Formula I think is that it shows an equivalence between an algebraic function and a trigonometric function.
(Note that I'm only a mildly learned laymen when it comes to mathematics. Any experts in math should feel very free to tell me why I'm wrong.)
As others have said, it is beautiful because of the relationship between e and pi. This is mind blowing. What do these 2 constants have in common? Not a thing. I feel the same way about the pi / 4 = 3/4 x 5/4 x .... infinite series. Why a relationship between pi and prime numbers?
Given notation as a tool of thought, it is really about which notation you are most familiar with. No doubt some people would find lisp notation enlightening, and some people would find the following beautiful.
The relationship is the same regardless of the notation.