It's known that the set of meromorphic functions (functions to $\mathbb{C}\cup\{\infty\}$) on a complex variety $X$ forms a field, called the function field of $X$.
Edit: Thanks to the comment by @Qiaochu Yuan, I realize that $\mathbb{PC}^1(X)\neq$ the set of meromorphic functions, since we don't allow the constant function with constant value $\infty$ to be meromorphic.
So is it possible that, as a scheme over $\mathbb{C}$, the projective line $\mathbb{P}^1_\mathbb{C}$ is a group scheme?
What if $\mathbb{C}$ is replaced by a general field $K$ or even $\mathbb{Z}$?
(I have never seen an explicit projective group scheme of the form $\mathop{\mathrm{Proj}}S$ in any reference, i.e. stating the multiplication morphism. All explicit examples introducing the group schemes are the affine ones. Hence stating any projective group scheme of the form $\mathop{\mathrm{Proj}}S$ while stating the multiplication morphism in the comment or answer will be nice)
No. A stronger statement is true: $S^2 \cong \mathbb{CP}^1$ is not a topological group. There are several different ways to see this and you can check out e.g. this math.SE answer for details. Here it would suffice to prove the weaker claim that $S^2$ is not a Lie group, which can be done using the hairy ball theorem or by classifying $2$-dimensional Lie algebras. A general fact which implies this conclusion at-a-glance is that if a compact triangulable path-connected space admits a topological group structure then by the Lefschetz fixed point theorem it must have Euler characteristic $0$ (or be a point).
Edit: Actually for the desired statement it suffices to show that $\mathbb{CP}^1$ is not a complex Lie group which is even easier and can be done by classifying $1$-dimensional complex Lie algebras. There's only one, namely $\mathbb{C}$, so $\mathbb{C}$ is the unique simply connected $1$-dimensional complex Lie group.
Examples of projective group schemes are given by elliptic curves and more generally by abelian varieties. The group operation is a bit annoying to write down explicitly but very classical. Somewhat surprisingly they are all abelian, hence the name; see e.g. Lemma 39.9.5 in the Stacks Project.
In the complex analytic category elliptic curves are quotients $\mathbb{C}/\Gamma$ where $\Gamma$ is a lattice in $\mathbb{C}$, as they would have to be to be compatible with the above-mentioned argument about $1$-dimensional complex Lie groups and Lie algebras.