Are $i,j,k$ commutative? I am trying to understand quaternions. I read that Hamilton came up with the great equation:
A) $i^2 = j^2 = k^2 = ijk = −1$
In this equation I understand that $i,j,k$ are complex numbers.
Later on, I read that 
B) $ij=k$
C) $ji=-k$
So, if $i,j,k$ are complex numbers, and complex number multiplication is commutative, why are these two equations different? I do understand that quaternion multiplication is non-commutative, but I do not understand why multiplying these complex components are also non-commutative.
Could someone please help me understand what is going on here. I (obviously) am not an expert in mathematics, so a simple explanation would be greatly appreciated.

Indeed commutativity still holds in the quaternions if your complex number only contains i's, j's or k's. I.e. any number's of the form $a + bi + 0j + 0k$ are commutative to each other, similarly $a + 0i + bj + 0k$ is commutative with other numbers of that form also and same with $a + 0i + 0j + bk$. However commutativity doesn't hold over the entire group of quaternions equipped with multiplication as its operation. $\mathbb{Q}_8$ is a non-abelian group in other words.
