We know $(a+b)^2=a^2+b^2+2ab$ if $a,b\in\mathbb R$.
Does that and other algebra identities hold if a,b belongs to complex numbers?
Given $a,b,c\in \mathbb{C}$, $$\begin{align}|a + b - c|^2 + |b + c - a|^2 + |c + a - b|^2 &= 12\\ |2a|^2 + |2b|^2 + |2c|^2 &= 12 - |a + b + c|^2\end{align}$$ using the identity $$(a + b + c)^2 + (a + b - c)^2 + (a + c - b)^2 + (b + c - a)^2=(2a)^2 + (2b)^2 + (2c)^2.$$
Is the above step correct?
In the complex numbers $|a|^2$ is not always equal to $a^2$. For example $|i|^2 = 1$ but $i^2=-1$.
So the deduction is not correct. The identity $(a+b+c) ^2 + (a+b-c) ^2 +(a+c-b) ^2 +(b+c-a) ^2 = (2a)^2 + (2b)^2 +(2c)^2$ will still hold however.