Another charming equation is the solution to i to the power of i. Apparently there are many or an infinite number, but the most natural or simplest is: e^(-pi/2), which is real, and something like:
0.207879576

You might be interested in an extension of the complex numbers (hypercomplex numbers) in which there are an infinite number of different numbers whose square is equal to -1.

ie. Is, e^(pi)(i)=e^(pi)(j) ? Is, i^j=j^i ?

For example: (a+bi+cj+dk) = (a+bi)+(c+di)j.
These complex-complex numbers are commutative, associative and distributive wrt addition.

By definition: i^2=j^2=-1, ij=ji=k, ik=ki=-j, jk=kj=-i, k^2=+1,
~(i=j), ~(i=k), ~(j=k), ~(1=i), ~(1=j), ~(1=k).

Note: these 4-D hypercomplex numbers are not 'quaternions', and they have most of the properties of complex numbers (eg. elementary functions) unlike quaternions.
The most noteworthy exception is the appearance of 'zero divisors'.
That is, (1+k)(1-k)=0 but neither term is equal to 0.
Another example: (i+j)(i-j)=0.

Division by 0 or a zero-divisor is not permitted.

Unlimited extensions of these (field-like) hypercomplex numbers are available.