In Conway's "On Numbers And Games," page 44, he writes:

NON-STANDARD ANALYSIS

We can of course use the Field of all numbers, or rather various small subfields of it, as a vehicle for the techniques of non-standard analysis developed by Abraham Robinson. Thus for instance for any reasonable function $f$, we can define the derivative of $f$ at the real number $x$ to be the closest real number to the quotient

$$\frac{f[x + (1/\omega)] - f(x)}{1/\omega}$$

The reason is that any totally ordered real-closed field is a model for the elementary states about the real numbers. But for precisely this reason, there is little point in using subfields of $\mathbf{No}$ when so many more visible fields will do. So we can say in fact the field $\mathbf{No}$ is really irrelevant to non-standard analysis.

Conway seems to make clear that you could, if you wanted, use the surreal numbers for non-standard analysis. This is because they are a real-closed field and thus a model for the theory of all elementary (first-order) statements about the reals. Conway does also make the point, in the last two sentences, that he doesn't view nonstandard analysis as the ultimate application of the surreals. But, for the sake of curiosity, I'm quite interested in understanding how you could do what Conway is hinting at above.

The statement that you could use the surreals for nonstandard analysis would seem to be much stronger than just saying it's a real-closed field. For instance, the real algebraic numbers are real-closed, but that doesn't mean that for some real algebraic $x$, $\sin(x)$ is also a real algebraic number. But we do have the corresponding result for the hyperreals, which is why we can transfer functions like $\sin$ to begin with to take nonstandard derivatives. So Conway would appear to be asserting an analogous "transfer principle" exists in some form that would let some general extension of $f(x + 1/\omega)$ for arbitrary $f$ to even exist for the surreals at all, in his expression above. (Or if he doesn't mean this I don't know how else to interpret it.)

So the main question is: how would such a transfer principle work? Even in a wishy-washy non-constructible sense. If you're using an ultrapower of the reals, everything is rather straightforward: every hyperreal is a set of reals (or an equivalence class thereof), and to transfer any first-order predicate to some hyperreal, you simply ask the predicate of every real in the set and see if it's true of "most" of them (where "most" means "in the ultrafilter"). Thus you have a real-closed field, a "transfer principle," and all of that. But is there some way to get a similar result using the language of the surreals, doing something clever with birthdays, left and right sets, and so on?

Some additional details:

There has been a little bit of prior discussion about this but not quite as directly as this question is asking. But for instance see here, where it is talked about that the surreals do have a kind of transfer principle in that they are isomorphic to the proper-class sized ultrapower of the reals (with many possible isomorphisms), and thus each isomorphism can be thought of as (unconstructably) defining a corresponding way to transfer first-order properties of reals on the surreal numbers. But also this transfer principle isn't quite as "natural" in the sense that there doesn't seem to be any "standard" choice of isomorphism. The punchline of all of this was, even though there's this isomorphism in theory, there is no canonical way to say what $\sin(\omega)$ is, for instance, or even $\omega \mod 2$.

On the other hand, though, it doesn't seem that this is any different from the way the hyperreals work. With those, we have the same problem, which is that all of the ultrafilters are typically non-constructive. And for instance, just like there is some debate about how the $\sin$ function should transfer to the surreals, or what $\sin(\omega)$ should be, there is also no real clear answer regarding what $\sin((1,2,3,4,...))$ should be, where $(1,2,3,4...)$ is a particular hyperreal number - because the answer depends on the ultrafilter, which determines what $(1,2,3,4...)$ even means to begin with, or what properties it has. So it doesn't really seem like the surreals are any better or worse off than the hyperreals in this regard.

And yet Conway is making the claim that he is. So there is this subtlety here regarding what really constitutes a "transfer principle" that I am hoping to get cleared up. Basically, with the hyperreals, part of the appeal is that even though you usually can't quite hold an ultrafilter in the palm of your hand the way you'd like, you can get a good bit of the way there constructively. You know, for instance, that whatever $\sin((1,2,3,4,...))$ is, in some sense it's $(\sin(1), \sin(2), \sin(3), \sin(4), ...)$. Since we don't know what the ultrafilter is we don't know exactly what properties that has, but we do know something: we know that for any ultrafilter this value will not be an integer, for instance. We can also "partly construct" an ultrafilter by constructing one that consistently assigns a membership value to all of the definable subsets of indices, and leave the undefinable ones open in a sort of choose-your-own-adventure approach, which is related to what Terry Tao calls "cheap nonstandard analysis". And then there are certain models of set theory which really do have constructible ultrafilters.

Summarizing, it seems implicit in Conway's assertion that you should also at least be able to do this kind of thing, that is to get "part of the way there" - at least have enough "transfer" for us to be sure that the expression $f(1 + 1/\omega)$ even makes sense to begin with. Perhaps part of my question is not quite so formal, but more in the direction of if it's possible to similarly get "part of the way there" but directly using the language of birthdays, left and right sets, etc, and all that stuff that makes the surreal numbers interesting, rather than sort of cheating and just using this isomorphism from the hyperreals. At least that seems like what Conway is suggesting, as it seems that it wasn't until later work by Keisler and then Ehrlich that it was proven that the proper class sized ultrapower of the reals and the surreals are isomorphic, if I interpret the above link correctly, but Conway clearly had something in mind even prior to that.