Extended complex numbers: what kind of algebra is it? We know that the complex numbers form a field. But when we include $\infty$, zero gains a multiplicative inverse. What is this new number system?
 A: The proper name for the complex numbers with $\infty$ adjoined is the extended complex plane or extended complex numbers. Since the operations are not completely defined for the whole set, it is not really an algebraic structure in itself, but the mappings of the structure onto itself are of extreme interest. 
Sometimes people use this interchangably with Riemann sphere, but IMO that connotes extra structure beyond merely the adjunction of $\infty$. If one imbues $\mathbb C$ and $\mathbb C\cup\{\infty\}$ with topology, then one can also speak of the one-point compactification of $\mathbb C$.

zero gains a multiplicative inverse.

No, it doesn't, not if you use the phrase "number system" in the normal sense$^\ast$, that is, there are additive inverses and multiplication distributes over addition. In that case, $0\cdot \infty=(1-1)\cdot\infty=\infty-\infty=0$, so $0\cdot \infty\neq 1$, as would be needed to qualify as a multiplicative inverse. The problem is that $\infty$ has no additive inverse.
On the other hand, you can relax what you mean by "number system" and call it something. I am not familiar with the "wheel" structure I see mentioned in ziggurism's answer, and I will defer to that to explain that type of number system.
$^\ast$Classically I think number system even referred to just fields...
A: As other answers point out, the traditional algebraic structures of rings or fields do not allow you to include a multiplicative inverse of $0$. The existence of such an element would violate the ring axioms, except in the most degenerate cases. So if you want you can instead understand $1/\infty=0$ in the real or complex projective line topologically, as a statement about limits.
However there is another algebraic structure that axiomatizes and allows to reason about division by zero, which is called a wheel. Any commutative ring may be embedded in a wheel. The algebraic structures of the real and complex projective line are examples of wheels (as long as you also include an element for $0/0$). 
You asked about algebraic structure, but for the sake of completeness I will also record that the complex projective line is topologically a 2-sphere, the same thing as the 1-point compactification of the plane at $\infty$, or endowed with a complex structure it is the Riemann sphere, that other answers have mentioned. The real projective line is similarly a circle, the one point compactification of the line $\mathbb{R}\cup\{\infty\}.$
A: Joining $\infty$ to the complex plane "compactifies" the space i.e,
$$\mathbb{C} \cup \{ \infty\}$$
 is the one point compactification of the complex plane. The ambients space you recover resembles $S^2$, but it now has a natural complex structure. We refer to it as the Riemann Sphere.
A: You can add a point at infinity to the complex plane to make a $2$-sphere. That's essentially a topological construction. Then you can say that $1/z$ has value $0$ at $\infty$, as a statement about limits. But there's no way to do arithmetic with $\infty$ that preserves the usual rules of arithmetic (that is, the field axioms).
