Let $a,b,c,d \in \mathbb R$ such that $c\ge 1$ and $0 < d < 2$. Let $i$ be the imaginary unit and $\Gamma$ the Euler Gamma.
Numerically it looks that
$$2^{(a+bi)}\Gamma\left(\frac{c-(a+bi)}{d} \right) \Gamma\left(\frac{(a+bi)}{2}\right) \asymp 2^{\frac{2(a+bi)}{d}} \Gamma\left(\frac{c-(a+bi)}{2}\right)\Gamma\left( c\left(\frac12 - \frac1d \right)+\frac {(a+bi)}{d}\right)$$
where $\asymp$ means bounded from above and below up to a constant (maybe depending on $a$ and $b$).
Can one prove this analytically?