Euler's formula is a specific case of the exponential map from Lie theory. This means e^x can be used with all sorts of interesting x types, and it often has surprisingly intuitive behavior! When x is a real number you get continuous growth. When x is a purely imaginary number you get continuous rotation. When x is complex you get continuous growth and rotation. When x is a matrix you get a continuous linear transformation (growth, rotation, and shear). What's the similarity here? Euler's formula treats it's argument as a transformation which gets continuously applied in infinitesimal amounts. This also explains the formula for calculating the value of e:

    e = lim (1 + 1 / n) ^ n where (x -> infinity)

https://en.wikipedia.org/wiki/Matrix_exponential

https://www.youtube.com/watch?v=O85OWBJ2ayo