Is $SU(n)$ a complex Lie group?

Question

I had been naively thinking that $GL_n(\mathbb C)$,$SL_n(\mathbb C)$,$U_n(\mathbb C)$ and $SU_n(\mathbb C)$ being the complex analogs of $GL_n(\mathbb R)$ etc should be complex Lie groups.

But wiki says that while $GL_n(\mathbb C)$ is indeed a complex Lie group, $SU_n(\mathbb C)$ is not.

1. What makes the general linear group a complex Lie group but the special unitary group not one?

2. What can be said about the special linear and unitary group over $\mathbb C$?