Everywhere I'd looked for an explanation of this angle-adding phenomenon, it seemed to have been in one of two forms:
Either something roughly like this:
$$\left(\cos\left(a\right)+i\sin\left(a\right)\right)\left(\cos\left(b\right)+i\sin\left(b\right)\right)=$$ $$\cos\left(a\right)\cos\left(b\right)+i\cos\left(a\right)\sin\left(b\right)+i\sin\left(a\right)\cos\left(b\right)-\sin\left(a\right)\sin\left(b\right)=$$ $$\cos\left(a+b\right)+i\sin\left(a+b\right)$$
or something to do with: $$e^{i\theta}=\cos\left(\theta\right)+i\sin\left(\theta\right)$$ These are fine, but they seem like shots in the dark that happened to hit.
What I mean (for the first one at least) is that it appears that multiplying sines and cosines where one of the coefficients squares to -1 just happens to fall into the proper form for the angle addition formulas, it's not intuitive that such would be the case.
I was simply wondering if there was an explanation that would make it seem 'obvious' that complex numbers should add their angles through multiplication, not just an explanation that shows that they do.
To my mind the most conceptual way is to go in the other direction, and define the complex numbers in terms of rotations.
This can be done as follows: consider the set of all linear transformations $T : \mathbb{R}^2 \to \mathbb{R}^2$ which scale and rotate the plane. This is equivalent to asking that $T$ is a similarity that fixes the origin, where a similarity is a map that multiplies all distances between vectors by the same number. This set of transformations is clearly closed under composition; more specifically, if two transformations $T_1, T_2$ scale by $r_1, r_2$ and rotate by $\theta_1, \theta_2$, then their composite $T_1 \circ T_2$ scales by $r_1 r_2$ and rotates by $\theta_1 + \theta_2$.
The elements of $T$ can be described more explicitly as follows. Suppose $T \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] = \left[ \begin{array}{c} a \\ b \end{array} \right]$. This already uniquely determines $T$, since it means that $T$ must be a scaling by $r$ and a rotation by $\theta$ where $(a, b) = (r \cos \theta, r \sin \theta)$. It's not hard to see geometrically that this implies that $T \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] = \left[ \begin{array}{c} -b \\ a \end{array} \right]$ (draw a diagram), so it follows that
$$T = \left[ \begin{array}{cc} a & -b \\ b & a \end{array} \right].$$
Somewhat surprisingly, this implies that the collection of similarities is closed under addition; this is specific to the case of $\mathbb{R}^2$ and is false in higher dimensions. This means it is in fact a real subalgebra of $M_2(\mathbb{R})$, and it is isomorphic to the complex numbers via the map
$$\left[ \begin{array}{cc} a & -b \\ b & a \end{array} \right] \mapsto a + bi \in \mathbb{C}.$$
From this point of view $i$ corresponds to the $90^{\circ}$ rotation $\left[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right]$, so $i^2 = -1$ just says that two $90^{\circ}$ rotations make a $180^{\circ}$ rotation. And Euler's formula is a special case of the matrix exponential, describing the infinitesimal generator of a rotation. Multiplication corresponding to addition of angles is built into the definition here and the surprise is that the complex numbers are closed under addition.