Why do we teach complex numbers?

Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics.

after algebra II, they never use complex numbers until pretty much complex analysis.

I assume you mean "they never use complex numbers in a math course until..." Many of them will use complex numbers in an engineering or physics course, and they will simply never take complex analysis.

I'm also not convinced that STEM students will never see a complex number in a math course until complex analysis. For example, in freshman calculus they may learn to do an integral like $\int e^x \sin x dx$ using complex numbers. Complex numbers come up in linear algebra as eigenvalues.

When do laymen use complex numbers in real world applications?

I'm confused by your usage of "laymen" here. When I first read the question, I took this to mean non-mathematicians (i.e., 99.999% of students in Algebra II). But in your comment on Daniel R. Collins's answer, you seem to be talking about "laymen" as including bartenders and truck drivers...? Does "laymen" include engineers? The majority of the population never understands or uses any math beyond arithmetic (even if they are forced to go through the motions of taking a course such as algebra). The reason we force college-bound students to take a course like Algebra II is that it indicates that they have the ability to do abstract thought.

We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has precisely n roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of course, this statement can only be made in the language of complex numbers.

Complex numbers are the answer to what's the algebraic completion of polynomial roots?

Complex numbers serve as an introduction to higher-degree metric spaces and matrices (i.e., point the way towards linear algebra).

Complex numbers are used in many real-world applications (esp. any case where a 2-dimensional measurement is neatly packaged thus): Control theory, fluid dynamics (flow in two dimensions), electrical engineering (impedence), signal analysis, fractals, etc. (https://en.wikipedia.org/wiki/Complex_number#Applications).

In the Preface to his Visual Complex Analysis, Tristan Needham writes:

If one believes in the ultimate unity of mathematics and physics, as I do, then a very strong case for the necessity of complex numbers can be built on their apparently fundamental role in the quantum mechanical laws governing matter. Also, the work of Sir Roger Penrose has shown (with increasing force) that complex numbers play an equally central role in the relativistic laws governing the structure of space-time. Indeed, if the laws of matter and space-time are ever to be reconciled, then it seems very likely that it will be through the auspices of the complex numbers.

I am not a mathematician, nor a teacher, but I can give an example of how learning complex numbers in school made me fall in love with mathematics.

It was the first time i realized it was possible to be creative with numbers. "What if we just define sqrt(-1) to be something". I remember realising that I could have thought of that. It wasn't hard, it wasn't some professor somewhere thinking up rules, i could have tried that. Up until this point, math had always been a set of rules that were supplied that could never be broken. And this one made me realize that these rules can sometimes be questioned and broken. I could try something and see what happens. It could be a creative, experimental, process.

Later on, i also realised that certain " what ifs" can make calculations easier. Sometimes the r-theta space makes the numbers easier than the x,y space. Sometimes the laplace transform is where its at. Sometimes using complex numbers is easier than not using them. Someone, somewhere, said "what if i try this... " to create the mathematical tools i use today as a Engineer.

Complex numbers is an example of how a very simple " what if" can create a new way of answering previously unanswered questions. It is an example of how math can be a creative experimental thing.

I think your premise is flawed "Why do we teach students about complex numbers if most will never reach a course that uses them?"

As soon they get a calculator they will test $sqrt(-1)$. They are going to see and answer and ask you what it means.

Also, " When do laymen use complex numbers in real world applications?" Check out this answer.