The concept of the imaginary golden ratio is fairly well known and it is described here and has appeared in an MSE question here Imaginary Golden Ratio.
The point of the imaginary golden ratio is that is derived from equations like those for the golden ratio and, despite being complex, shares many of the attributes of the golden ratio.
Consider these three paths to the golden ratio, $\varphi$ and its imaginary counterpart, call it $\varphi_i$.
$$ \begin{align} &\text{Ratios}\\ &\varphi=\frac{a}{b}=\frac{a+b}{a}\\ &\varphi_i=\frac{a}{b}=\frac{a-b}{a}\\ \end{align} $$
$$ \begin{align} &\text{Quadratic equations}\\ &\varphi^2-\varphi-1=0\\ &\varphi_i^2-\varphi_i+1=0\\ \end{align} $$
$$ \begin{align} &\text{Recurrence relations}\\ &f_n=f_{n-1}+f_{n-2}\\ &f_n=f_{n-1}-f_{n-2}\\ \end{align} $$
The imaginary golden ratio so obtained is $\varphi_i=\frac{1+\sqrt{3}i}{2}=e^{i\pi/3}$
(We already know that $ f_n=-f_{n-1}+f_{n-2}$ has the characteristic roots $[1/\varphi,-\varphi]$ and leads to the negafibonacci numbers found here.)
We set out to explore an alternative imaginary golden ratio, call it $\varphi_j$, based upon
$$ \begin{align} &\varphi_j=\frac{a}{b}=-\frac{a+b}{a}\\ &\varphi_j^2+\varphi_j+1=0\\ &f_n=-f_{n-1}-f_{n-2}\\ \end{align} $$
Indeed, we found $\varphi_j=\frac{-1+\sqrt{3}i}{2}=e^{i2\pi/3}$, which is similar to the original but has its own unique properties.
All the golden ratios have a relation with their inversions, to wit,
$$ \varphi=1+\frac{1}{\varphi},\quad \varphi_i=1-\frac{1}{\varphi_i },\quad \varphi_j=-1-\frac{1}{\varphi_j } $$
where $\psi=-1/\varphi$ and $\psi_{i,j}=1/\varphi_{i,j}$.
All of the recurrence relations satisfy the Binet-like equation (for $f_{0,1}=[0,1]$ as with the Fibonacci sequence)
$$ f_n=\frac{\varphi_k^n-\psi_k^n}{\varphi_k -\psi_k} $$
where $\psi=-1/\varphi,\ \psi_{i,j}=1/\varphi_{i,j}$.
Of course, the recurrence relations for the imaginary golden ratios are periodic but differ from each other as shown here.
$$ \varphi_i: f=\{0,-1,-1,\dots \text{repeats} \}\\ \varphi_j: f=\{0,-1,+1,\dots \text{repeats} \}\\ $$
Like the golden ratio, these complex counterparts can be expressed as continued fractions,
$$ \begin{align} &\varphi=[1;1,1,1,\dots]=\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots}}}\\ &\varphi_i=[1;-1,-1,-1,\dots]\\ &\varphi_j=[-1;-1,-1,-1,\dots]\\ \end{align} $$
and continued square roots (or nested radicals)
$$ \begin{align} &\varphi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}\\ &\varphi_i=\sqrt{-1+\sqrt{-1+\sqrt{-1+\sqrt{-1+\dots}}}}\\ &\varphi_j=\sqrt{1+i\sqrt{1+i\sqrt{1+i\sqrt{1+\dots}}}}\\ \end{align} $$
So, even though $\varphi_j=\varphi_i^2$, they are decidedly different, and we think it warrants introducing another imaginary golden ratio. In addition, there are many interesting interrelationships between the two imaginary ratios. For example,
$$ \begin{align} &\varphi_j^2+\varphi_j+1=0 \Rightarrow \varphi_i^4+\varphi_i^2+1=0\\ &-\varphi_j=\varphi_i^5= \varphi_j^{5/2}= \varphi_j^3 \varphi_j^{-1/2}=\varphi_j^{-1/2} \end{align} $$
These relations and several others can become apparent when looking at $\sqrt[3]{1}$ and $\sqrt[6]{1}$ on the unit circle, as shown in the figure below.
I am seeking external validation for this analysis.
