How do complex numbers apply in real life?
A few thoughts...
When you travel in a straight line, it makes sense to measure the distance you travel in the form of a single real number.
When considering rotations, then we often measure the angle as a single real number too, but what if you make one or more complete revolutions? You may not be interested in the total angle turned through, but just the resulting difference between the starting and finishing orientations. If that is so, then it may make more sense to think of the angle as a complex number of modulus
More generally, any affine transformation in
Scaling: Multiplication by a real quantity.
Rotation: Multiplication by a value on the unit circle.
Reflection: Complex conjugate.
So many everyday circumstances where basic
Affine transformations are used extensively in computer games and general animation.
If you have an application of trigonometric functions, then you are using something which is more natural in terms of complex numbers. For example an sinusoidal curve looks a little contrived as a shape, whereas a circle seems much more natural.
In fact we find that trigonometric, hyperbolic, exponential and logarithmic functions are all members of one family of functions. What we see when we look at these as real valued functions of real values is an artificial slice of a richer whole - the shape of these functions as complex valued functions of complex numbers. It's just a little difficult for us to visualise four dimensional space.