In the context of drug experiences "hyperspace" is arguably a redundant term so long as it is analogous with "breakthrough". I think it was originally coined to describe DMT experiences where hyperspatial perception (beyond 3D) occurs. It's not something that happens often with DMT (and I've never heard it reported outside of DMT experiences), but there are nevertheless many reports of it. For instance,
this one (at the top Erowid notes it's 5-MeO-DMT and LSD, however I think this is a classification mistake as 5-MeO is never mentioned in the text, only DMT).
Here's a quote:
i am impossible geometry
being schlanged
by impossible geometry.
i am the hypercube-4D-tesseract
being fucked by a klein bottle
inside of a sideways figure eight
I think unfamiliarity with the spatial connotations of hyperspace in the drug community resulted in a lot of people hearing it and assuming it was a synonym of "breakthrough experience". "Hyperspace" is arguably more precisely used, then, as a subtype of breakthrough experience. A breakthrough experience describes a massive, most often sudden, qualitative shift in the experience of a drug. It is qualitatively different than being at a standard plus +++, but is not necessarily a plus ++++ (technically, a breakthrough could happen at a + or ++ as well, but usually happens at high doses). On this understanding, Shulgin's description of a plus ++++ is also a subtype of breakthrough experience (a massive qualitative shift to a sense of love and universal unity). Even the term breakthrough is not always used this way to describe drug experiences -- someone might say they broke through to the fourth plateau on DXM, even though the forth plateau is supposed to be an experience consistently accessible purely as a function of dose -- but I think it's the closest single term we have to get at the concept.
I've experienced hyperspace only twice, with both times involving a mix of DMT and salvia. The first time I experienced it I basically mumbled "impossible" over and over for about 5 minutes because I never imagined a perceptual system evolved and matured in three-dimensional space could ever produce a genuine extra-dimensional perception (it's impossible for me to imagine now, but during it the recognition was overwhelming, even having known little about extra-space at the time). I do believe hyperspace to be the work our perceptual systems under special highly exotic conditions, just the same, and not an independently existing realm (this is not to say extra-spatial realities don't exist, just that I don't believe our minds are "transported" to them by drugs).
One of the more complete discussions having to do with independently existing realms (hyperspatial or not) being accessed by drugs can be found in the thread
Salvia, the Conveyor Belt & Other Common Themes. The essential rationale for the belief that an independent realm is accessed by salvia is the fascinating overlap between surprisingly specific trip themes between different users. I remain skeptical, but the confluence of reports between salvia users is the best evidence I've seen yet.
