Solution:
We have the equation:
$$x^{3} + x^{2} + x + 1 = 0.$$

We can factor the left side:
$$x^{3} + x^{2} + x + 1 = (x^{3} + 1) + (x^{2} + 1) - x^{2} - 1 + x^{2} + 1 = (x^{3} + 1) + (x + 1).$$
But it's easier to factor by grouping:
$$x^{3} + x^{2} + x + 1 = (x^{3} + x^{2}) + (x + 1) = x^{2}(x + 1) + 1(x + 1) = (x^{2} + 1)(x + 1).$$
So,
$$(x^{2} + 1)(x + 1) = 0.$$
Therefore, the solutions are the roots of $x^{2} + 1 = 0$ and $x + 1 = 0$.

1. $x + 1 = 0 \implies x = -1$

2. $x^{2} + 1 = 0 \implies x^{2} = -1 \implies x = i$ or $x = -i$

Thus, all complex solutions are:
$$x = -1,\ i,\ -i.$$