Question: I need to solve this complex equation:
$$
(c + id)^2 = 1
$$

where $i^2=-1$.

What am I supposed to calculate here? Just $c$ and $d$?

Answer: 
From the fundamental theorem of algebra, you know that the equation $z^2=1$, a quadratic one, has at most two roots.

One of the roots is the obvious $z=1$, and with a few seconds of thinking, $z=-1$ is the other.
By the binomial formulas we know that 
$$
0=z^2-1=(z-1)(z+1)
$$
has the only solutions $z=\pm 1$. This is true in the real numbers and stays true in the field of complex numbers.