Let $p$ be a prime of type $3\space mod \space 4$. Then there is no solution $x^2 = -1 \space mod \space p$. Therefore we can define the so-called Galois imaginary $i$. ( $i^2 = -1 \space mod \space p$ )
Now by adding this galois imaginary , we consider $a + bi \space mod \space p$.
Computing with this $a+bi$ is similar to usual modular arithmetic. We know for instance how to simplify $(a+bi)^c$ , multiply , add and substract.
However , does it make sense to consider
$(a + bi)^{c + di} \space mod \space p$
where $a,b,c,d$ are integers and $d$ is nonzero ?