  Addressing a problem posed by W. Li and A. Wei (2009), we investigate the
average number of (complex) zeros of a random harmonic polynomial $p(z) +
\overline{q(z)}$ sampled from the Kac ensemble, i.e., where the coefficients
are independent identically distributed centered complex Gaussian random
variables. We establish a precise asymptotic, showing that when $\text{deg} p =
\text{deg} q = n$ tends to infinity the average number of zeros is asymptotic
to $\frac{1}{2} n \log n$. We further consider the average number of zeros
restricted to various regions in the complex plane leading to interesting
comparisons with the classically studied case of analytic Kac polynomials. We
also consider deterministic extremal problems for harmonic polynomials with
coefficient constraints; using an indirect probabilistic method we show the
existence of harmonic polynomials with unimodular coefficients having at least
$\frac{2}{\pi} n \log n + O(n)$ zeros. We conclude with a list of open
problems.
