This is not a directly a matter of hyperbolic geometry but of complex Euclidean geometry. The construction of "impossible" triangles is the same as the construction of square roots of negative numbers, when considering the coordinates the vertices of those triangles must have. If you calculate the coordinates of a triangle with sides 1,3,5 or 3,4,8 you get complex numbers. In ordinary real-coordinate Euclidean geometry this means there is no such triangle. If complex coordinates are permitted, the triangle exists, but not all its points are visible in drawings that only represent the real points.
In plane analytic geometry where the Cartesian coordinates are allowed to be complex, concepts of point,line, circle, squared distance, dot-product, and (with suitable definitions) angle and cosine can be interpreted using the same formulas. This semantics extends the visible (real-coordinate) Euclidean geometry to one where any two circles intersect, but possibly at points with complex coordinates. We "see" only the subset of points with real coordinates, but the construction that builds a triangle with given distances between the sides continues to work smoothly, and some formulations of the law of Cosines will continue to hold.
There are certainly relations of this picture to hyperbolic geometry. One is that $cos(z)=cosh(iz)$ so you can see the hyperbolic cosine and cosine as the same once complex coordinates are permitted. Another is that the Pythagorean metric on the complex plane, considered as a 4-dimensional real space, is of the form $x^2 + y^2 - w^2 - u^2$, so that the locus of complex points at distance $0$ from the origin contains copies of the hyperboloid model of hyperbolic geometry. But there is no embedding of the hyperbolic plane as a linear subspace of the complex Euclidean plane, so we don't get from this an easier way of thinking about hyperbolic geometry.
To help visualize what is going on it is illuminating to calculate the coordinates of a triangle with sides 3,4,8 or other impossible case, and the dot-products of the vectors involved.