[gunnihinn]

For any smooth function (like the conjugate) it makes sense to ask whether there exist holomorphic functions that approximate it arbitrarily well.

However: Suppose that for every n > 0 there exists a holomorphic function f_n such that |f_n(z) - z| < 1/n for all z. Then |f_n(z)| <= |f_n(z) - z| + |z*| = |z| + 1/n by the triangle inequality. A consequence of Liouville's theorem is that any entire holomorphic function with polynomial growth is a polynomial; here in particular we would need to have f_n(z) = a_n z + b_n for some complex numbers a_n and b_n. For real x we would have |(a_n - 1)x + b_n| < 1/n for all x, so a_n = 1. For imaginary iy we would have |(a_n + 1)iy + b_n| < 1/n for all y, so a_n = -1, which is a contradiction.

In fact, if a sequence of holomorphic functions converges uniformely on compact sets, the limit is itself holomorphic because of Cauchy's theorem.