I see the answer I gave yesterday as a guest is not posted, an email check revealed I did not properly register...

Ok, now being a registered thing I can say:

The Cauchy-Riemann equations directly relate to all entire functions being harmonic. In $\mathbb{C}$ you can even factorize the Laplacian operator into two things and you see that only one of those factors slams the stuff to zero so the outcome is harmonic.

Basically the CR-equations say:

$$
\frac{\partial}{\partial y} = i \cdot \frac{\partial}{\partial x}
$$
so that
$$
\frac{\partial^2}{\partial y^2} = - \frac{\partial^2}{\partial x^2}
$$
Hence all analytic (entire) functions on $\mathbb{C}$ are also harmonic but it does not go the other way...

Now for $\mathbb{R^3}$ space there is indeed a complex multiplication possible but this does not give rise to harmonic functions. In this answer it will go too far to explain how 3D complex multiplication works, but as far as I know after 26 years of looking at that stuff:

You cannot make harmonic functions that way.

Here comes a small monkey out of the sleeve:
I have a separate website related to all 3D complex numbers things:

http://3dcomplexnumbers.net/

The 3D Cauchy-Riemann equations are beautiful yet they do not give harmonic stuff. Hope I answered your question a tiny bit...