I considered simplifying $\sqrt {1+\sqrt 5}$.

So I started $(a+b \sqrt 5)^2 = 1 + \sqrt 5$.

This gave me $a\color{blue}{^2} + 5b^2 =1 , 2ab = 1$ so the result was

$ \sqrt{1+\sqrt 5} = \sqrt{1/2 - i} + \dfrac{\sqrt 5}{\sqrt{2 - 4i}}.$

I continued in the same spirit : $(a\_2+b\_2 i)^2 = 1/2 - i. $

$a\_2^2 - b\_2^2 = 1/2, 2a\_2b\_2 = -1.$ This gave me :

$ \sqrt{1+\sqrt 5} = \dfrac{\sqrt{1+\sqrt 5}}{2} - \frac{1}{4} (\sqrt 5 - 1)(\sqrt{1+\sqrt 5})\color{blue}i +\dfrac{\sqrt 5}{\sqrt{2 - 4i}}\tag1$ 

I was a bit surprised by the self-similarity.

I got the feeling of " periodic " or " fractal " , like running in circles.

It this normal to get this kind of self-similarity when the initial radical cannot be rewritten in simpler shorter form ( without complex numbers or nesting ) ?

Are they ways to compute or theorems to know about just how many steps it takes to get the self-similarity ?

Does this imply that the number of nontrivial non-nested expressions for a real radical is Always finite ?

***edit***

Perhaps this edit clarifies a bit what I meant.

Basicly the only identities and substitutions for identities we can have are :

$ \sqrt{1+\sqrt 5} = \sqrt{1/2 - i} + \sqrt{1/2 + i}$

$ \sqrt{1/2+ i} = \dfrac{\sqrt{1+\sqrt 5}}{2} - \frac{1}{4} (\sqrt 5 - 1)(\sqrt{1+\sqrt 5})i$

$ \sqrt{1/2- i} = \dfrac{\sqrt{1+\sqrt 5}}{2} + \frac{1}{4} (\sqrt 5 - 1)(\sqrt{1+\sqrt 5})i$

And no more than those 3. Or 2 if we do not count conjugates as seperate.

Now I suspect this mysterious number 2 is no coincidence ;

***Conjecture***

If a nested root form containing only sqrts , positive integers and additions (thus no divisions or substractions ) is given that cannot be simplified within the same ring then the amount of identities ( as in the example above , ignoring conjugates ) is at most the amount of sqrt's in the expression.

In this case 2 sqrt's => 2 identities.

***edit***

***comment***

A similar thing happens with $\sqrt{5+\sqrt 3}$. You end up with a $\sqrt{10+2\sqrt 22}$ term and when you go on you end up with $\sqrt{5+\sqrt 3}$ again.

Here $\sqrt{10+2\sqrt 22}$ takes on the " role " of $\sqrt{1/2 + i}$ but this examples demonstrates that the phenomenon can stay within the reals too.

These examples are fairly easy to check and in that sense they may appear trivial , but there must be more behind this. 
The same thing happens for cuberoots for example , and that gives us a mimimum polynomial of degree 9 which is much less well understood then degree 4.

Similar things happen if we use Bring radicals.

Although intuitive and easy to check any particular example by basic algebra , group and ring theory ... The basic framework is unclear to me.
I do not think Galois theory alone explains it completely.

I gave a bounty but without succes.

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