Imaginary but real

$i^i=e^{i \ln i}=e^{i\times \frac{i\pi}{2}}=e^{\frac{-\pi}{2}}=0.20787957635$ (Since $i=e^{\frac{i\pi}{2}}\implies \ln i=i\frac{\pi}{2}$ ) $\Large \therefore \lfloor 10000\times i^i\rfloor=\lfloor2078.8\rfloor=\boxed{2078}$

$\huge \lfloor\: 10000\: \cdot \:(i^{i}) \rfloor =$

$\huge \lfloor 10000\: \cdot \:((e^\frac{i\pi}{2})^{i}) \rfloor =$

$\huge \lfloor 10000\: \cdot \:(e^\frac{-\pi}{2}) \rfloor =$

$\huge 2078$