Well that's the thing. They're not unconnected. The hype depends on the people who aren't aware of the connection.
1's origin is in counting.
pi's origin is in the study of the circle.
0's origin is in accounting.
i's origin is polynomial algebra.
e's origin is in the study of exponentials and derivatives.
The relationships between all of these fields is a discovery in itself, and a revelation that was not apparent at all throughout history, and it is still not apparent to most people who haven't studied math in-depth.
Basically, e^ipi + 1 = 0 is proof of the connection, one that is not at all immediately apparent, and has not been known for the majority of humanity's history.
Let D_n be a difference operator like D_n(F) = F(x+n) - F(x)
Solving by induction for the equation D_n(F)/D_n(x) = F(x), F(0)=1, it gives (1+n)^(x/n).
So now you have a "family" of functions that when "derived" by its difference, gives the same constant ^ x. Euler is the case when n→0.
This aplies for every n, including complex numbers.