generalisation of Cauchy-Riemann equations to 3D Hi, harmonicity in 2d is preserved under mappings that satisfy the Cauchy-Riemann equations.
What about 3D? What conditions should a mapping satisfy to preserve harmonicity?
is there a general characterization a la CR for 3D?
Here is an example of non-trivial such mapping
Let $u(x,y,z)=U(X,Y,Z)$ where $$X=xy+z,~~~~ Y= \frac{\sqrt3}4 (x^2-y^2) - \frac{xy}{2}+z
,~~~~ Z= -\frac{\sqrt3}4 (x^2-y^2) - \frac{xy}{2}+z$$
(I found this example by first assuming $X=xy+z$ then guessing for Y,Z from the overdetermined system that they satisfy... hope it's right...)
 A: See 
MR0545705 (80k:58045) 
Ishihara, Tôru
A mapping of Riemannian manifolds which preserves harmonic functions. 
J. Math. Kyoto Univ. 19 (1979), no. 2, 215–229. 
58E20 (31C12) 
The author shows that a mapping has the title property in dimension at least three if and only if it is either constant or a Riemannian covering up to homothety.. For $\mathbb{R}^n$ this means a composition of translations, rotations, and homothety ($x\rightarrow a x,$ for some $a>0.$)
A: Your "examples" usually don't work. Up to a complex conjugation (= symmetry), a $2$D-mapping $\phi$ that preserves harmonicity is a holomorphic function, via $z=x_1+ix_2$ and $f(z)=\phi_1+i\phi_2$. Now, let us form $\psi(x_1,x_2,x_3)=(\phi(x_1,x_2),x_3)$. What happens is that
$$(\partial_1^2+\partial_2^2)(u\circ\phi)=|f'|^2(\partial_1^2u+\partial_2^2u)\circ\phi.$$
From this, you see that $\Delta(u\circ\psi)$ is not proportional to $(\Delta u)\circ\psi$ if $|f'|\ne1$ (notice that if $|f'|\equiv1$, then $f$ is an affine isometry). Hence $\psi$ does not preserve harmonicity. More generally, a function $\phi$ preserves harmonicity ($u$ harmonic implies $u\circ \phi$ is harmonic) if and only if ${\rm D}\phi(x)$ is a similitude, that is the product $\rho(x)R_x$ of some isometry and of a homothety. 
On the contrary, a theorem due to Liouville says that a mapping in ${\mathbb R}^3$ that preserves angles must be an affine similitude. This applies in particular to mappings that preserve harmonicity. Edit (after comments below): Liouville's Theorem actually says that a direct $C^4$ conformal map is the composition of an affine similarity and possibly of inversions $x\mapsto a\|x-x_0\|^{-2}(x-x_0)$.
A: 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... 
A: Actually, the correct notion of maps preserving harmonicity is that of 'harmonic morphisms':  A map $f:(M,g)\to (N,h)$ between Riemannian manifolds is a harmonic morphism if it pulls back $h$-harmonic functions on $N$ to $g$-harmonic functions on $M$.  There are many, many nontrivial examples, and there is a large literature on the subject.
There is an extensive Atlas of Harmonic Morphisms (see http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/atlas.html) that contains a useful bibliography.
