A second timelike dimension would have to be folded up to very small size like the other extra dimensions. If the second timelike dimension were similar to the first in size, just orthogonal, it would make too many things possible. If all things are possible, then evolution is dead and change is impossible, so there is a phenomenological contradiction. OTOH it would eliminate the need for the Big Bang.
Indeed, the compactification of extra spatial dimensions is an artificial construct.
For instance, under ekpyrotic cosmology, the inception dimensions are deemed parallel and thus, there is no big bang.
Likewise, in 5 dimension space/time such compactification is not necessary:
If our time dimension is a plane and not a line, the past, present and future are all seen at once and the cause/effect can be inverted to effect/cause.
Therefore, IMHO the artificial aversion of science is a prejudice which may stand between us and truth.