From the Oxford English Dictionary (subscription only)
The noun dimension
is first recorded in the English language a little in 1413
1413 Pilgr. Sowle (1483) v. xiv. 107 Ther is no body parfit withouten thre dymensions, that is breede, lengthe, and depnesse.
It took another ~450 years for dimension to be applied to time:
1858 W. Whewell Hist. Sci. Ideas ii. viii. §4–5 Time is conceived as a quantity of one dimension..Indeed the analogy between time, and space of one dimension, is so close, that the same terms are applied to both ideas.
and a few years more to use it in the sense of "other dimensions" in the sense we now use it.
1878 B. Stewart & P. G. Tait Unseen Universe vii. §220. 221 Suppose our (essentially three-dimensional) matter to be the mere skin or boundary of an Unseen whose matter has four dimensions.
These all seem to be secular references though. I suspect that "God in another dimension" arose around the late 1920s with the eccentric [wiki]John G. Bennett[/wiki], who
suggested a six-dimensional Universe with the usual three spatial dimensions and three time-like dimensions that he called time, eternity and hyparxis. Time is the sequential chronological time that we are familiar with. The hypertime dimensions called eternity and hyparxis are said to have distinctive properties of their own. Eternity could be considered cosmological time or timeless time. Hyparxis is supposed to be characterised as an ableness-to-be and may be more noticeable in the realm of quantum processes. [Edit GB: or not.]
While Bennett has put forward some curious speculation, his ideas stop at some subjective aspects of the perception of time which does not place them on a fully scientific basis. The question of measurement, how one would measure these hypothetical extra time-like dimensions, is left unaddressed.
Bennett is perhaps best known for his many books on psychology and spirituality, and particularly the teachings of G.I. Gurdjieff, and starting the British section of the [wiki]Subud[/wiki] movement.
The basis of Subud is a spiritual exercise commonly referred to as the 'latihan kejiwaan', which was said by Muhammad Subuh to represent guidance from "the Power of God" or "the Great Life Force".
Add to this dumbed down science, superman and other sci-fi comic strips in the 1950s or 60s and these elements seem to account for the common low level of critical thought among Christians and others of a religious bent; they are determined to use big words to make their delusions sound important.