As others have pointed out, every complex number (except $0$) has two distinct square roots. There's an old-fashioned expression: "multiple-valued function". The idea is that the square root function has two values rather than just one: the square root of $4$ is "$\pm2$". The term is simply a misnomer according to the definition of "function" that's been used for probably almost a century now, and I know of at least one professor who was quite offended by the expression for that reason.
But there's also the idea of "branches". Suppose $z=1$ so that $\sqrt{z}=1$. Then let $z$ move in the positive direction (counterclockwise) around the unit circle centered at $0$. As $z$ moves around the circle, $\sqrt{z}$ moves around the circle half as fast. By the time $z$ reaches $-1$, $\sqrt{z}$ has moved half as far along the circle and reached $i$. $z$ keeps going, so that by the time $z$ has gone all the way around the circle, $\sqrt{z}$ has moved half-way around and reached $-1$. There you have a square root of $1$, and it is $-1$. As $z$ continues to move around the circle, $\sqrt{z}$ continues to move half as fast, and when $z$ reaches $-1$ for the second time, $\sqrt{z}$ reaches $-i$. So that's the second square root of $-1$. $z$ then keeps going, and when it reaches $1$ for the second time, $\sqrt{z}$ then comes back to $1$. So one says that this function has two "branches". When you return to $1$ for the first time, you're in the other branch than the one you start in. But when you return to $1$ again, you're back in the same branch you started in. In the same way, the cube-root function has three branches. And some functions have infinitely many branches. The arctangent function is one of those.