Can there be other multiplications on $\mathbb{R}^2$ making it a ring? Together with addition $+ : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$
$$(x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1 + y_2)$$
the multiplication $\cdot : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$
$$ r_1 e ^{i\phi_1} \cdot r_2 e ^{i\phi_2} = (r_1 r_2)e^{i(\phi_1 + \phi_2)}$$
make the plane $\mathbb{R}^2$ a ring. (Please forgive me making use of $\mathbb{C}$-notation, I guess it's clear what I mean.)
I wonder if there may be other multiplications – i.e. functions $\times : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$ – which together with $+$ make the plane $\mathbb{R}^2$ a ring.

Yes, there are. One is the component-wise multiplication:
$$
(x_1, y_1)\cdot (x_2, y_2) = (x_1x_2, y_1y_2)
$$
Another (if you don't require rings to have a multiplicative unit) is the trivial multiplication
$$
(x_1, y_1)\cdot (x_2, y_2) = (0,0)
$$
and I am sure there are many others.
As a side-note, you could have something like
$$
(x_1, y_1)\cdot (x_2, y_2) = (x_1x_2, x_1y_2 + x_2y_1 + 2y_1y_2)
$$
which is actually the complex numnbers again, only camouflaged: $(1, 0) = 1$ and $(0,1) = 1+i$.
