Per convention, we draw a 2d Cartesian coordinates system as a '+' with the x-axis pointing right and the y-axis pointing up.
When it comes down to making the plot 3d, by adding a z-axis, it's pretty common to rotate the plot and represent it with z up, x pointing to our left side and y pointing to our right side.
Couldn't we just leave x pointing to the right, y pointing up and z kind of being the dimension that projects above the plot?
I assume this is just how things are, but normally conventions in math are based on a simple way of doing things. In this case it seems to add an unnecessary complication, or is this rotation useful?
There is a very strong mental association between a varying quantity and the vertical dimension. It’s there even in the language we use to describe the variation: when a value increases, we speak of its “going up;” when it decreases, we speak of its “going down.” Thus, we conventionally plot the graph of a single-variable function in two dimensions with the independent variable (the abscissa) varying horizontally and the dependent variable (the ordinate) varying vertically. For a function of two variables, it then seems natural to keep the vertical axis for the dependent variable, and add another axis in the horizontal plane for the second independent variable. This also matches our everyday experience in that we’re largely free to move along the surface of the earth as we like—the independent variables—but we’re pretty much constrained to follow the ups and downs of that surface as we move around—the dependent variable.
We get some clashing conventions here, though. In 2-d, we label the abscissa $x$ and the ordinate $y$. If you think of a 3-d graph as extending this with another axis, we might’ve kept the labels of the original two axes and labeled the new axis $z$. Unfortunately, the convention for functions of two variables is to call the independent variables $x$ and $y$ and the dependent one $z$, so from this point of view we end up relabeling the axes as well.
There’s also a preference for right-handed coordinates systems, which is why the positive directions of the $x$- $y$- and $z$-axes are arranged according to the right-hand rule.
None of these conventions are universal. In 2-d computer graphics it was common practice to use a left-handed coordinate system, with $x$ increasing toward the right, but $y$ increasing downward on the screen/window/viewport. This is a very text-oriented point of view: in all of the languages in use today that are written horizontally, successive lines of text proceed down the page. Similarly, in languages that are written vertically, the text proceeds from top to bottom.
In 3-d graphics and computer vision, it’s common practice to have the camera’s axis coincide with the $z$-axis, so that a $z$-value corresponds to the depth of an object in the scene. Moreover, if the camera is facing in the negative direction, then in the standard position the image and scene $x$- and $y$-axis directions coincide, resulting in a particularly simple form for the standard camera matrix. Curiously, in those sources that conventionally point the camera in the positive $z$-direction, a right-handed coordinate systems are still used for both the image and scene, necessitating some minus signs in the transformation matrix and providing endless opportunities for students to make silly sign errors.