This is related to this question. I took the parabola $x^2$ and rotated it 45 degrees clockwise using a parametric equation for the parabola --- $x(t) = t, y(t) = t^2.$ After rotating, I get the the points $$(t \sqrt{2}/2 + t^2 \sqrt{2}/2, -t \sqrt{2}/2 + t^2 \sqrt{2}/2).$$
On the usual parabola $x^2$, centered at $x = 0,$ I know I can get an addition operation: $$P_1 + P_2 = (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, (x_1 + x_2)^2),$$ which is obviously a point on the parabola, so we have closure.
If I put this addition through the 45-degree rotation, I expect to also get closure for addition on the points of the rotated parabola. I get the formula $P_1 + P_2 = (x_1, y_1) + (x_2, y_2)$ =
$$( (x_1 + x_2)\sqrt{2}/2 + (x_1 + x_2)^2 \sqrt{2}/2, -(x_1 + x_2)\sqrt{2}/2 + (x_1 + x_2)^2 \sqrt{2}/2)$$
I've checked my calculations by adding and plotting the addition-results on the parabola. The additions do land on the parabola, so I'm confident we get closure.
There's one thing that is strange to me. All additions end up falling on the lower side of the rotated parabola. For instance, $$E + F = (0, \sqrt{2}) + (\sqrt{2}, 3\sqrt{2}) = (1 + \sqrt{2}, -1 + \sqrt{2}).$$
This addition is (I hope) the same as the geometric addition usually used to define addition of points on a parabola. We take a straight line $S$ through $E, F,$ take a parallel $T$ to $S$ passing through the origin (0,0) and define the sum $E + F$ to be the intersection of $T$ and the parabola. There are two intersections: (0,0) and the point $E + F.$ The geometric procedure gives me a point on the upper side of the parabola. The addition formula defined above gives me a point on the lower side of the parabola. (I'm always getting additions on the lower side of the parabola, so something must be wrong with the addition formula, but I don't know what.)
Also, once I get this formula right, I'm going to wish for a way to reduce the points of the parabola modulo a prime $p$, but I'm already puzzled as to how I could do that if I have $\sqrt{2}$ in every coordinate.
Note that the group is just $(\mathbb{R},+)$, i.e. addition on the real numbers.
I think that the formula on the rotated parabola you're looking for is: $$(x_1 \sqrt{2} / 2 +x_1^2 \sqrt{2} / 2, -x_1 \sqrt{2} / 2 +x_1^2 \sqrt{2} / 2 ) + (x_2 \sqrt{2} / 2 +x_2^2 \sqrt{2} / 2, -x_2 \sqrt{2} / 2 +x_2^2 \sqrt{2} / 2 ) = ((x_1+x_2) \sqrt{2} / 2 +(x_1+x_2)^2 \sqrt{2} / 2, -(x_1+x_2) \sqrt{2} / 2 +(x_1+x_2)^2 \sqrt{2} / 2 )$$
About the geometric definition. Say you want to add $(x,x^2)$ and $(y,y^2)$. Let's assume $x \neq \pm y$. Then you're looking for a point $(z,z^2)$ such that the slope of the line through $(0,0)$ and $(z,z^2)$ is same as the slope of the line through $(x,x^2)$ and $(y,y^2)$. I.e. $$\frac{z^2}{z} = \frac{x^2 -y^2}{x-y}$$ This simplifies to $$z= x+y$$ which is again just the ordinary addition of real numbers.
(If $x=y$, replace the line through the summands by the tangent to the parabola $P$ at $(x,x^2) = (y,y^2)$ and apply the same argument. If $x= -y$, $(x,x^2)+(-x,x^2)=(0,0)$. Alternatively, use that the addition as defined is continuous and $\{((x,x^2),(y,y^2))| x \neq \pm y \}$ is a dense subset of $P^2$.)