Conway showed that the Field of surreal numbers ("${\bf No}$") is the maximal totally ordered Field.

Later Jacob Lurie showed that the Group of all partizan games ${\bf Pg}$ is the universally embedding partially ordered Abelian Group.

Is there some analogous functorial characterization of the Field of surcomplex numbers ${\bf No}[i]$?

Or might there be some sense in which ${\bf No}[i]$ isn't the "right" algebraic closure of ${\bf No}$? (Recall what happens when one takes the algebraic closure of the field of $p$-adic numbers: one gets a system that is unsatisfactory because it is not metrically complete, and then one has to pass to an even larger system to obtain the correct $p$-adic analogue of the field of complex numbers. Of course this is a vague analogy; in particular, the notion of metric completeness is not relevant in the case of ${\bf No}[i]$.)

Come to think of it, why is ${\bf No}[i]$ algebraically closed?