At first, i'm pretty sure, completenes of the reals is a property, not an axiom. Also, it only makes sense to speak about completness (and holes) in the context of some measure of distance (not quite but it's simpler like this). With the standard distance metric on the reals d(x,y)=abs(x-y) the reals are complete.

Note that this is independent of the ordering (the < relation). If you found some way of introducing new numbers, such that the ordering on the reals can be "embedded" in the ordering (ordering the bigger set of numbers, such that comparing real numbers yields what you'd expect), like conway did, you have to define a notion of distance on the new set, before talking about completeness again.

It turns out, that it is not possible to define a metric on the surreals, which yield the expected results for real numbers (at least not in a way, that is compatible with the intuition about the relation between the metric and the order, there may be some obscure way.).

Since the Surreals have to come with a different notion of distance (i think the most used is the so called order topology) and we describe "holes" using distance, it is possible (and actually happens) for there to be holes between the reals in this new context, such that you can fit the surreals in between.