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This is just out of curiosity, but as I see many nice results in introductory complex analysis course, which mostly are from viewing $\mathbb{C}$ as a field structure and $\mathbb{R}^{2}$ and using topology of it, I started to wonder if people study geometry on $\mathbb{R}^{3}$ with many copies of complex planes (3 trivial ones are $xy, yz, xz$-planes). I will appreciate nice reference.

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For 3D space I'd rather look towards quaternions than complex numbers. See en.wikipedia.org/wiki/… –  MvG Oct 19 '12 at 21:40
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I believe that one of the reasons complex analysis has such wonderful results is due to the beautiful interplay between the algebraic structure of $\mathbb{C}$ as a complete, algebraically closed field, and the topology it has when viewed as a two-dimensional euclidean space. Unfortunately, things start to break down when you try to extend things to $\mathbb{R}^3$ or use quaternions in $\mathbb{R}^4$.

It is possible to get a multiplication in $\mathbb{R}^3$ by using the vector product, but it fails to be associative, so you don't get anything as nice as a field, and with the quaternions in $\mathbb{R}^4$ you lose commutativity, so again you don't get a field.

In total, it seems that everything just works so very neatly in $\mathbb{C}$, that any attempt to extend the number of dimensions has to break something, unless you want to go down the route of several complex variables, and do function theory $\mathbb{C}^n$, but that is whole new story ...

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One way to do this is to look at what is called the mini-twistor space of $\mathbb{R}^3$. This is the set of all oriented straight lines in $\mathbb{R}^3$ and has a natural complex structure. It is naturally isomorphic to the tangent bundle of the two-sphere $TS^2$. It's not hard to see that because every oriented straight line in $\mathbb{R}^3$ is determined uniquely by a unit vector $u$, in the direction of orientation, and a shortest vector $v$ from the origin to the line. These must zero inner product because $v$ is the shortest vector to the line. This construction lets you relate certain linear and non-linear differential equations on $\mathbb{R}^3$ to complex analytic objects on the mini-twistor space like sheaf cohomology groups and algebraic curves.

You can find more in the twistor theory literature or in Nigel Hitchin's paper

N.J. Hitchin. Monopoles and geodesics. Comm. Math. Phys., 83(4):579--602, 1982.

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In electromagnetism, Gauss' law says

$$ \frac{q}{\varepsilon_0} = \oint \mathbf{E}\cdot d\mathbf{A}$$

this is kind of like Cauchy residue theorem in 3D. Maybe you could say:

$$ \int_{\mathbb{R}^2} \frac{dx\, dy}{(1 + x^2 + y^2)^{3/2}} = 2\pi$$

by placing a single charge at $(0,0,1)$ above the complex plane and integrating over a hemisphere of radius $R \to \infty$ based in the xy-plane.

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Similarly, people might be interested in this. (I can email the PDF if anyone doesn't have access through a university.) –  AndrewG Jan 1 '13 at 11:49
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