I do not think I appreciated the formula until I had been exposed to the exponential map in its generalized forms: The so-called 'matrix exponential' and the exponential map of Lie algebra. They place Euler's humble formula into a grander and rather beautiful setting.
Now, I like to think of exponentiation as a kind of integral over infinitesimal generators; and $i$ just happens to be a generator for rotation about a circle in the plane (aka $\mathrm{U}(1)$ aka $x \mapsto e^{it}x$).
Now, I like to think of exponentiation as a kind of integral over infinitesimal generators; and $i$ just happens to be a generator for rotation about a circle in the plane (aka $\mathrm{U}(1)$ aka $x \mapsto e^{it}x$).