I have that f(z) = u(x,y) + iv(x,y) is analytic.

Under what circumstances will g(z) = u(x,y) - iv(x,y) be analytic?

Using the Cauchy-Riemann Equations,

du/dx = dv/dy and du/dy = -dv/dx

and for g to be analytic,

du/dx = -dv/dy and du/dy = dv/dx

but these equations put together imply,

dv/dy = -dv/dy, du/dy = -du/dy, etc.

It seems to me that u(x,y) and v(x,y) must be constants.

Is this true?

Note all derivatives are partial derivatives obviously.