Is it possible that a polynomial of degree $n$ with real coefficients has exactly one complex root? I saw https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem but wondered if this can happen for polynomials several variables.
What do you mean with complex root? If you consider real numbers as particular complex numbers ($i.e.$ complex numbers with zero-imaginary part) then every polynomial of the form $p(z)=(z-z_0)^n$ where $z_0$ is a real number has exactly one complex root.
But if you mean a complex number which is not real; then this is impossible. Indeed, let $z_0=a+ib$ be any complex number; then the only polynomials which have $z_0$ as unique complex solution are $p(z)=(z-z_0)^n$ where $n$ is a non-zero natural number. But clearly, in this case, your polynomial has not real coefficients.
This is technically possible if you consider the root of a real valued line is a one real number, and all real numbers are complex.
Edit: this doesn't violate your theorem since any real number is also it's complex conjugate