Can This Definition of Pi Work in More Abstract Algebras?

[Gerenuk]

I quite like the following definition of pi and I wonder which minimal algebraic rules are needed to make this definition work?

\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n=1
\therefore |a|=2\pi k

(For example, are there algebras more general than complex numbers, where this works?)

 

[g_edgar]

Well, in the quaternions e^{2\pi u} = 1 where u is any quaternion with u^2 = -1

 

[Gerenuk]

So quaternions probably also work.

Any more abstract algebra? Hmm, what do I need? I need addition, multiplication, scalar division, limiting process and modulus, right? What has to be fulfilled as to yield pi then?