In this question, somebody asked about a homogeneous but nonlinear function.
The answer in general form was:
"Actually the most general function with the desired property would be $f(x)=\alpha x+\beta\left|x\right|$ with $\alpha, \beta \in\mathbb{R}$ arbitrary real constants."
How can this be generalized into multiple dimensions?
My gut instinct is that this indirectly enables you to define a linear function for each orthant, which is just the positive and negative side for this one-dimensional case.
Fortunately there is a simple characterisation of positive homogeneous, not necessarily linear functions. Unfortunately, these functions don't match your intuition in general, and are quite nasty to work with (and nastier to plot). Basically, it comes down to the following fact:
In other words, homogeneousness is determined on rays beginning at the origin. Think about it: the condition $f(\alpha x) = \alpha f(x)$ relates the function values on two points on a single ray, as $x$ and $\alpha x$ are vectors pointing in the same direction. This condition has no bearing on any other ray! There's no way to relate $f(x)$ to $f(y)$ where $x$ and $y$ are linearly independent. Even if $x$ and $y$ point in opposite directions, we are still unable to relate $f(x)$ and $f(y)$.
So, this means, we can describe the positive homogeneous functions $f$ entirely by their restriction to the unit sphere $S$ in $\Bbb{R}^n$. Indeed, we can take any function $g : S \to \Bbb{R}$, and obtain a unique positive homogeneous function $f$ such that $f|_S = g$. And this function $g$ has no restrictions; it could be totally discontinuous, for example.