As a monoid, \$$\langle \mathbb{C}, +, 0 \rangle\$$ is isomorphic to \$$\langle \mathbb{R}^2, +, (0, 0) \rangle\$$, which has a natural partial order given by the product on \$$\langle \mathbb{R}, \le \rangle\$$ -- in other words, we have \$$(x, y) \le (x', y')\$$ iff \$$x \le x' \land y \le y'\$$. The monoid structure respects this order, since if \$$x \le x'\$$ and \$$r\$$ is any real number, we definitely have \$$x + r \le x' + r\$$ and \$$r + x \le r + x'\$$; this lifts to the product straightforwardly. We can carry this order across the isomorphism back to the complex numbers, giving a monoidal poset structure on \$$\mathbb{C}\$$.

This is much like Maria's construction in #6, but -- if I understand correctly -- we avoid identifying numbers whose components have the same magnitude, e.g. \$$1 + i\$$ and \$$-1 - i\$$.

We can incorporate Matthew's ideas as true posets here: since rotations about the origin are automorphisms of the complex plane, we can similarly rotate our partial order. For instance, a 90-degree rotation would give us a poset where \$$-1 + i \le -2 + 2i\$$. For every line of the complex plane passing through the origin, we can obtain a rotation that maps this line to the \$$x\$$-axis; every such line gives another induced partial order.