Given points A(9, 1), B(4, 13), C(-8, 8), and D(-3, -4) in the complex plane, prove that ABCD is a square.

The length of a segment between two complex numbers $z_1$ and $z_2$ is given by $|z_2 - z_1|$.

Length of AB:

$$|AB| = |z_B - z_A| = |(4 + j13) - (9 + j)| = |(4 - 9) + j(13 - 1)| = |-5 + j12|$$ $$|AB| = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$

Length of BC:

$$|BC| = |z_C - z_B| = |(-8 + j8) - (4 + j13)| = |(-8 - 4) + j(8 - 13)| = |-12 - j5|$$ $$|BC| = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13$$

Length of CD:

$$|CD| = |z_D - z_C| = |(-3 - j4) - (-8 + j8)| = |(-3 - (-8)) + j(-4 - 8)| = |5 - j12|$$ $$|CD| = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$

Length of DA:

$$|DA| = |z_A - z_D| = |(9 + j) - (-3 - j4)| = |(9 - (-3)) + j(1 - (-4))| = |12 + j5|$$ $$|DA| = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13$$

Since $|AB| = |BC| = |CD| = |DA| = 13$, all four sides are equal. This means ABCD is a rhombus.

We can determine the angle between two adjacent sides, for example, $\vec{AB}$ and $\vec{BC}$, by examining the ratio of the complex numbers representing these vectors.
The vector $\vec{AB}$ is represented by $z_B - z_A = -5 + j12$.
The vector $\vec{BC}$ is represented by $z_C - z_B = -12 - j5$.

Consider the ratio $\frac{z_C - z_B}{z_B - z_A}$:

$$\frac{z_C - z_B}{z_B - z_A} = \frac{-12 - j5}{-5 + j12}$$

Multiply the numerator and denominator by the conjugate of the denominator:

$$\frac{-12 - j5}{-5 + j12} \times \frac{-5 - j12}{-5 - j12} = \frac{(-12)(-5) + (-12)(-j12) + (-j5)(-5) + (-j5)(-j12)}{(-5)^2 + (12)^2}$$ $$= \frac{60 + j144 + j25 + j^2 60}{25 + 144}$$ $$= \frac{60 + j169 - 60}{169}$$ $$= \frac{j169}{169} = j$$

Since the ratio $\frac{z_C - z_B}{z_B - z_A} = j$, this means that the vector $\vec{BC}$ is obtained by rotating the vector $\vec{AB}$ by $90^\circ$ counter-clockwise. Therefore, $\angle ABC = 90^\circ$.

Since all four sides of the quadrilateral ABCD are equal in length and one of its interior angles is $90^\circ$, ABCD is a square.

$\text{ABCD is a square.}$