Understanding complexity in multi-dimensional arrays can be challenging, but let's break it down into simpler parts.

First, let’s talk about space complexity. This term refers to how much memory (or space) we need to store data. For a two-dimensional array, which you can think of as a table with rows and columns, the space needed grows based on how many items are in it. If we have a table with $m$ rows and $n$ columns, we need space for $m \cdot n$ items. As we add more dimensions, the space needed can increase a lot. Figuring out how much memory we need can get tricky.

Next, we have access patterns. This is about how we get to the data in these arrays. Multi-dimensional arrays are more complex than one-dimensional arrays (which are like a single line of data). The layout of the data can change, depending on whether we store it row by row or column by column. This affects how long it takes to reach specific items. For example, to find an item in a 2D array using coordinates $(i, j)$, the time needed might differ based on how the memory is arranged, which complicates things.

Then, we must consider time complexity. This tells us how long a task will take based on the size of the data. When we multiply two $n \times n$ matrices (think of them as big tables), using a simple method can take a long time—about $O(n^3)$, which means if we double the size, the time needed increases quite a bit. However, smarter methods like Strassen's algorithm can make this faster, dropping the time to around $O(n^{2.81})$. Choosing the right method is important and can sometimes be confusing when explaining how fast things will run.

Lastly, we have edge cases. These are unusual situations that can pop up with multi-dimensional arrays. They can happen when the dimensions don’t match up well or include empty values. These odd scenarios make it harder to figure out general rules for understanding the complexity of an array.

In summary, looking at complexity in multi-dimensional arrays means dealing with space needs, different ways to access data, how long operations take, and tricky situations that can arise. Each of these parts is essential to truly grasp how well our data structures are performing.