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?