Uniquely determining complex multiplication I encountered the following problem and it's not entirely clear to me exactly what I am supposed to do:

Show that the following rules uniquely determine complex multiplication on $\mathbb{C}=\mathbb{R^2}$:
  
  
*
  
*(a) $(z_1+z_2)w=z_1w+z_2w$
  
*(b) $z_1z_2=z_2z_1$
  
*(c) $i\cdot i=-1$
  
*(d) $z_1(z_2z_3)=(z_1z_2)z_3$
  
*(e) If $z_1$ and $z_2$ are real, $z_1\cdot z_2$ is the usual product of real numbers.
  

I get that (a) refers to distributivity, (b) refers to commutativity, (c) I'm not sure what the goal is there (is there a special name for this property?), (d) refers to associativity, and (e) refers to something else. 
Looking at (b), for example (with $z_1=a_1+b_1i$ and $z_2=a_2+b_2i$), I get that
\begin{align}
z_1z_2 &= (a_1+b_1i)(a_2+b_2i)\\[0.5em]
&= (a_1a_2+a_1b_2i+b_1ia_2+b_1ib_2i)\\[0.5em]
&= (a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i
\end{align}
and
\begin{align}
z_2z_1 &= (a_2+b_2i)(a_1+b_1i)\\[0.5em]
&= (a_2a_1+a_2b_1i+b_2ia_1+b_2ib_1i)\\[0.5em]
&= (a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i
\end{align}
Since $\operatorname{Re}(z_1z_2)=\operatorname{Re}(z_2z_1)$ and $\operatorname{Im}(z_1z_2)=\operatorname{Im}(z_2z_1)$, it is clear that $z_1z_2=z_2z_1$. But what does this really show? It seems like the question is asking for more (something about uniqueness, etc.). 
Any ideas?

Well, using only the rules given, denoting the complex product with $\cdot$, we can say that for any complex numbers $z = a + i \cdot b$ and $w = c + i\cdot d$, we must have (using $(d)$ whenever there are terms involving at least two multiplications and writing $a \cdot b \cdot c$ for $a \cdot (b \cdot c) = (a \cdot b) \cdot c)$):
$$\begin{align*} z\cdot w &= (a+i\cdot b)\cdot (c+i\cdot d) \\ &= a \cdot (c+i\cdot d) + i\cdot b \cdot (c+i\cdot d) \quad \text{using (a)} \\ &= (c+i\cdot d) \cdot a + (c+i\cdot d)\cdot i\cdot b \quad \text{using (b)} \\ &= c\cdot a + i\cdot d \cdot a + c \cdot i\cdot b + i\cdot d \cdot i\cdot b \quad \text{using (a)} \\ &= c \cdot a + i \cdot d \cdot a + i \cdot c \cdot b + i \cdot i \cdot d \cdot b \quad \text{using (b)} \\ &= c \cdot a + i \cdot d \cdot a + i \cdot c \cdot b  - d \cdot b \quad \text{using (c)} \\ &= c \cdot a - d \cdot b + i \cdot (d \cdot a + c \cdot b)\quad \text{using (a)} \\
&= c a - d b + i \cdot (d a + c b)\quad \text{using (e), denoting real multiplication by juxtaposition} \\
\end{align*}$$
So complex multiplication is uniquely determined by these rules.
