Compact Hausdorff and C^*-algebra "objects" in a category. This is yet more on "algebraic objects in functional analysis".
Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable category representing functors from that category to CompHaus.
An obvious such functor is the spectrum of a unital $C^*$-algebra.  This seems to imply that $\mathbb{C}$ is a compact Hausdorff object in the category of unital $C^*$-algebras.  So:

Question 1: Is this right?

Followed by the obvious:

Question 2: Are there any other interesting "Compact Hausdorff" objects in other categories?

Similarly, $C^\ast$-algebras is algebraic, and whilst Banach spaces isn't algebraic then it embeds in an algebraic theory (of totally convex spaces).  Again, to any compact Hausdorff space one can assign its $C^\ast$-algebra of continuous functions to $\mathbb{C}$.  This suggests that $\mathbb{C}$ is a "$C^\ast$-algebra" object in CompHaus - except that $\mathbb{C}$ is not a compact Hausdorff space.  However, we have a way out due to the way that $C^\ast$-algebras are algebraic: it's the unit ball that we should be thinking of and this is continuous functions to the closed unit disc in $\mathbb{C}$, which is compact Hausdorff.  Thus $\{z \in \mathbb{C} : |z| \le 1\}$ seems to be a $C^\ast$-algebra object in Compact Hausdorff spaces.  Again:

Question 3: Is this right?

and

Question 4: Are there any other interesting "$C^\ast$-algebra" objects in other categories?

and

Question 5: Are there any "Banach space" objects (or "totally convex space" objects) floating around anywhere?

 A: This "answer" doesn't even get as far as answering question 1, but I'll go ahead anyway.  
All I want to say is how I think "compact Hausdorff space object" should be defined.  This should be equivalent to what Sridhar said, though I haven't stopped to think about it.  
Let $\mathcal{E}$ be a category with small products.  A compact Hausdorff object in $\mathcal{E}$ should be an object $X$ of $\mathcal{E}$ together with, for each set $I$ and ultrafilter $U$ on $I$, a function
\[
\xi_U: X^I \to X
\]
satisfying some axioms that I'm too lazy to write down, but will explain a bit in a moment.  
When $\mathcal{E} =$ Set, you can think of $\xi_U$ as specifying the $U$-limit of each $I$-indexed family of points of $X$.  (That there's exactly one limit point is the compact Hausdorff property.)  One axiom tells you what happens when $U$ is the principal ultrafilter on some $i \in I$: then $\xi_U$ sends a family x to $x_i$.  A second says something about limits of limits.  A third (and I think there are only three) says something about what happens when you have a map $I \to J$. 
This formulation doesn't come out of thin air, you won't be surprised to hear---there's a systematic process for taking a (suitable kind of) monad on Set and producing a definition of its "algebras" in any category with products.  But I won't go into that now.
A: Question 1: If I understand you correctly, you're proposing that $\mathbb{C}$ should be a compact Hausdorff object in some category because it represents a functor from that category to the category CH of compact Hausdorff spaces (in something like the sense that the functor $Hom(-, \mathbb{C})$ into Set factors through the forgetful functor from CH to Set). But I don't see why this should be sufficient to make $\mathbb{C}$ a compact Hausdorff object.
That is, presumably, from the approach of functorial semantics, a compact Hausdorff object in a category C should be a product-preserving functor from L to C, where L is the dual of the Kleisli category for the ultrafilter monad on Set (that is, L is the Lawvere theory whose category of (Set-)models is the category of compact Hausdorff spaces). I can see how, more generally, for any Lawvere theory L and category C, every C-model of L (i.e., a product-preserving functor F from L to C) induces a representable functor Hom(-, F(1)) from C to Set which factors through the forgetful functor from Set-models of L to Set. But it's not obvious to me that the converse of this holds as well (that every representable functor from C to Set with this factorization property arises from some C-model of L).
Perhaps I'm missing something and your reasoning for $\mathbb{C}$ being a compact Hausdorff object is something more than this. Perhaps I'm hopelessly confused. But, tentatively, I think the answer to question 1 is "No" or at least "Not necessarily".
(Edit: As seen below, the correspondence does go both ways, so the last line is retracted, leaving the second-to-last line...)
A: The "Bohrification" paper arXiv:0905.2275 may be relevant to Question 4. As I understand, they discuss the notion of $C^\ast$-algebra objects in a given topos.
