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: 
By far, the simplest solution is simply to take the square root of both sides:

$c+di = \pm 1$

which allows us to immediately equate real and imaginary parts to get $c = \pm 1, d = 0$.

Much simpler than squaring the LHS.
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.