I suppose it natural to write $f(x,y) = u(x,y) + iv(x,y)$ but it make me wonder are we losing anything in that process? When we talk about things component-wise, is that limiting anything?
Can we always assume that given any complex function $f(x,y)$ that it could be written $u(x,y)+iv(x,y)$? (yes, I think)
Can we always assume that we can find formulas/equations for $u$ and $v$? (no, I think)
Are there situations where even though we are given the equation for $f$, it is not possible to find equations for u and v - more substantial ones than $u=Re(f)$ and $v=Im(f)$ obviously ? (I'm not sure about this one but I suspect that it can never happen but then again...)
Anyone with anything to add is appreciated.