Undergrad Hypergeometric Limits: Analyzing p(x)

[thatboi]

I have been working with some Hypergeometric functions whose behavior I am not quite familiar with. Suppose the equation I wish to analyze is
$p(x) = (e^{x}-1)^{2i}\left({}_{2}F_{1}(a,b;c;e^{x}) + {}_{2}F_{1}(a+1,b+1;c+1;e^{x})\right)$ where $a,b,c$ are all complex valued and we have $\Re(c-a-b)>0$ and we have $\Re((c+1)-(a+1)-(b+1))=0$ but $(a+1)+(b+1)-(c+1) = 2i$. My question is, how do I take the proper limit of $p(x)$ as $x\rightarrow 0$. I believe the relevant identities are (15.4.20) and (15.4.22) from https://dlmf.nist.gov/15.4 but there is still the limit of figuring out what $0^{i}$ means.
Let me know if you guys have any ideas. Thanks!

 

[anuttarasammyak]

(e^x-1)^{2i}=e^{2\log(e^x-1)i}
So as x$\rightarrow$ 0, phase becomes undermined.