What is the interpretation of Fourier Transform containing only imaginary part?
The integral doesn't exist in the conventional sense. How did you evaluate the limit $\lim_{t\to\infty}e^{-j\omega t}$?

If one allows distributions (generalized functions), such as the Dirac impulse (and its derivatives), as Fourier transforms then the class of functions that can be transformed becomes much wider, and the unit step is one of the functions that has a Fourier transform in that sense. However, its transform is not given by the expression in your question. The Fourier transform of $u(t)$ is

$$\mathcal{F}\{u(t)\}=\pi\delta(\omega)+\frac{1}{j\omega}\tag{1}$$

Take a look at this answer for a derivation of $(1)$.

A simple sanity check also shows that the expression in your question cannot be the transform of $u(t)$. A purely imaginary Fourier transform corresponds to an odd time domain function, which $u(t)$ is not. The expression in your question is actually a valid Fourier transform, but of a different function. Its inverse transform is $x(t)=-\frac12\textrm{sgn}(t)$, which is obviously an odd function.