The golden egg

The value 0 is the set that is a distance 0 from its negative.
The unit diameter circle is the set that is a distance 1 from its negative.

The value 1 is the set that is a distance 0 from its inverse.
What is the set that is a distance 1 from its inverse?

One point in the set is the golden ratio (1.618..) since its inverse is 0.618... but it isn't the only point in the set, for instance 0.618 is also in the set. Also cos(30) + 0.5i is in the set since its inverse is cos(30) - 0.5i. The full set is drawn in blue:

The green curve is an ellipse, since the blue oval is 'fatter' at smaller real values and thinner at larger real values it is in egg shape; I'll call it the golden egg! In fact there are two of these, one being the negative of the other.

The left and right sides can be calculated iteratively and separately, for the left side (lesser real values):
a = 1/(c + a) where c is the unit radius circle.
For the right side:
a = 1/a - c

Its formula is:
|𝑧|⁴−|𝑧|²+1=2|𝑧|²cos(2arg(𝑧))  

Here's a better image:

We can extend this idea to the quaternions, giving a very similar looking shape:

but its a 3D surface in 4D, the above is a cross section at its largest point, but it is symmetric around the x (real) axis so it is simply a distorted 3-sphere in the manner shown.

Doesn't look much like an egg we're used to, but it is definitely an ovoid. If we scale down the non-x axes by two and colour it differently it looks more like your classic egg.

The width diameter is sqrt(5)/2 so with it scaled by half it is sqrt(5)/4. And its length is of course 1.