Question about quaternionic conjugation

Question

The quaternionic conjugation is defined by

$$\begin{aligned}i &\mapsto -i\\j&\mapsto -j\\k&\mapsto -k\end{aligned}$$

But since $ij=k$, shouldn't we have that $k = ij \mapsto (-i)(-j) = ij = k$?

Answer 1

That's because quaternion conjugation, unlike complex conjugation, is not an automorphism. In fact, for quaternions $a$, $b$ we have $$ (a b)^* = b^* a^* $$ so in you example $$ (ij)^* = j^* i^* = (-j)(-i)=ji=-ij=-k=k^* $$