Equation 17 p487 ....not the equation we're looking for

Thanks for the quick reply. You know, there is another reason why I really like the form $\partial_x + i\partial_y$, but it might be too far afield to be relevant here. Recently I was reading through some of Spivak's *Introduction to Differential Geometry*. In the discussion of the tangent bundle (Ch. 3, page 78 in my book), he shows, quite elegantly, that the space of all linear operators which are "derivations at $p$" is spanned by the familiar partial derivatives. Somehow, this rings a bell with the form of the equation with $\partial_x + i \partial_y$.

It also highlights a nice fact (I'm not sure where it is in the book...), that exposes a way of looking at complex multiplication. If $z = x + iy$, $\mathbb{z} = (x,y)$, $w = u + iv$, and $\mathbb{w} = (u,v)$, then

$$z \cdot w = \langle\mathbb{z},\mathbb{\bar w}\rangle - i\mathbb{z}\times\mathbb{\bar w}$$

While not as useful in general as considering complex multiplication to be a dilative rotation, it does at least bring complex multiplication into terms of familiar vector operations, and makes plausible (at least from a purely formal perspective) the formula

$$\partial_x + i\partial_y = \langle\nabla, \bar{\cdot}\rangle - i\nabla \times \,\bar{\cdot}$$

It may very well be that this is all balderdash, but I can't help but feeling there is some useful conceptual tool or somewhat interesting connection buried here somewhere...