[chombier]

Think of exponentiation of some number 'a' as in-between its integer powers: 'a^1.5' is kind-of half-way between 'a' and 'a^2'.

If you plot all the integer powers of 'a', they all belong to a curve and the exponential simply fills-in the gaps for non-integer exponents.

Now, there are many possible ways to fill the gaps but the exponential does it so that a^m * a^n = a^{m+n} holds even for non-integer numbers m and n.

Similarly, if you take integer powers of a complex number, they all lie on some curve and the exponential fills-in the gaps, again turning sums into products. The same works with matrices, and so on.