Suppose that $R$ is a relation on the set of complex numbers $\mathbb{C}$. The relation $R$ is defined as follows:
For any two complex numbers $w,z \in \mathbb{C}$, $$w R z \Leftrightarrow \operatorname{Re}(w)-\operatorname{Re}(z)=k, \quad \operatorname{Im}(w)-\operatorname{Im}(z) = l \sqrt{2}$$
for some integers $k,l \in \mathbb{Z}$.
Question: Describe the distinct equivalence classes of $R$.
How can I visualise the equivalence classes graphically?
So we know that $$w R z \iff \operatorname{Re}(w)-\operatorname{Re}(z)=k, \quad \operatorname{Im}(w)-\operatorname{Im}(z) = l \sqrt{2}$$
for some integers $k,l \in \mathbb{Z}$. This can be re-written as $$w R z \iff \operatorname{Re}(w)-\operatorname{Re}(z) \in \mathbb{Z}, \quad \operatorname{Im}(w)-\operatorname{Im}(z) \in \sqrt{2} \mathbb{Z}$$
Consolidating these into one expression, we have that \begin{align*} w R z & \iff w - z \in \mathbb{Z} + \sqrt{2} i \mathbb{Z} \\ \Rightarrow w R z & \iff w \in \mathbb{Z} + \sqrt{2} i \mathbb{Z} + z \\ \Rightarrow [z] & = \mathbb{Z} + \sqrt{2} i \mathbb{Z} + z . \end{align*}
Visually, this means that in the same way $\mathbb{Z}^{2}$ makes a "grid" (more specifically a lattice, as you don't have the sides of the rectangle, only the vertices) of $\mathbb{R}^{2}$, each of these equivalence classes makes a grid of the complex plane, where each equivalence class is a lattice that makes rectangles which are of length $1$ along the real axis, and length $\sqrt{2}$ along the imaginary axis. In other words, you can make the equivalence class $[z]$ by starting at $z$, gridding the plane with $1 \times \sqrt{2}$ rectangles (such that $z$ is a vertex of one of these rectangles), and then keeping only the vertices of the rectangles.
EDIT: In case you're unfamiliar, when given sets $A, B$, we define $A + B : = \{ a + b : a \in A, b \in B \}$. Similarly, we define $A + b : = \{ a + b : a \in A \}$.