[chombier]

For me what made it clicked is realizing that complex numbers are 2D matrices: z = x + i y = [[x -y][y x]].

So really we should be writing z = x * [[1 0][0 1]] + y * [[0 -1][1 0]], but since it's tedious we just call 1 == [[1 0] [0 1]] the 2x2 identity matrix and i == [[0 -1][1 0]], and check that i^2 = -1.

Then no more magical i number, the complex product can be derived from the matrix product, the exponential becomes the 2x2 matrix exponential, and so on.

[recursive]

Using a matrix as an exponent isn't any more comprehensible than an imaginary number to me. If it works for you, that's great, but it's not much help to me.

[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.