  A $d$-dimensional nowhere-zero $r$-flow on a graph $G$, an $(r,d)$-NZF from
now on, is a flow where the value on each edge is an element of $\mathbb{R}^d$
whose (Euclidean) norm lies in the interval $[1,r-1]$. Such a notion is a
natural generalization of the well-known concept of circular nowhere-zero
$r$-flow (i.e.\ $d=1$). For every bridgeless graph $G$, the $5$-flow Conjecture
claims that $\phi_1(G)\leq 5$, while a conjecture by Jain suggests that
$\phi_d(G)=1$, for all $d \geq 3$. Here, we address the problem of finding a
possible upper-bound also for the remaining case $d=2$. We show that, for all
bridgeless graphs, $\phi_2(G) \le 1 + \sqrt{5}$ and that the oriented $5$-cycle
double cover Conjecture implies $\phi_2(G)\leq \tau^2$, where $\tau$ is the
Golden Ratio. Moreover, we propose a geometric method to describe an
$(r,2)$-NZF of a cubic graph in a compact way, and we apply it in some
instances. Our results and some computational evidence suggest that $\tau^2$
could be a promising upper bound for the parameter $\phi_2(G)$ for an arbitrary
bridgeless graph $G$. We leave that as a relevant open problem which represents
an analogous of the $5$-flow Conjecture in the $2$-dimensional case (i.e.
complex case).
