The notion that Jesus didn't die on the cross is more commonly known as the swoon theory. It was originally presented during the 19th century, a time when many competing naturalistic explanations for the resurrection of Jesus were formed.
A well-esteemed resurrection researcher, Gary Habermas, describes this time as follows:
Another indication of the failure of the naturalistic theories is that each one was disproven by the 19th century liberals themselves. These scholars refuted each other's hypotheses, thereby leaving no viable alternative. For example, D. Strauss delivered the historical death blow to the swoon theory held by K. Venturini, H. Paulus and others. On the other hand, F. Schleiermacher and Paulus pointed out errors in Strauss' hallucination theory. However, the major decimation of the hallucination theory came at the hands of T. Keim. The fraud and legend theories were disproven by later critical research.
For a long time, the refutation of the swoon theory by Strauss kept the theory unsupported by specialists. It has seen only limited support:
The swoon theory has reappeared recently in a few places, although seldom among specialists. One of the only exceptions is the brief article by Margaret Lloyd Davies and Trevor A. Lloyd Davies. It develops the hypothesis that Jesus lost consciousness, causing the bystanders to conclude that he was dead. When taken down from the cross, he revived and was treated. Surprisingly, the appearances apparently seem not to be caused by Jesus actually being seen after his recovery, but by some unspecified sort of "perceptions," raising once again the issue of hallucinations. The medical outcry against the Davies' stance was instructive, with multiple reasons being given to indicate that Jesus really died by crucifixion.
So, long story short: there are very few supporters for the theory, especially among scholars. It's academically not plausible; you'll probably have better luck with some of the other naturalistic theories. See the refutation by Strauss for some problems with the theory.