Do "imaginary" and "complex" angles exist? During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting:
$ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $
$ \sin ( 1 + i ) \approx 1.30 + 0.63 i $
Does this mean that there is such a thing as "imaginary" or "complex" angles, and if they do, what practical use do they serve?

Here's a more literal interpretation.
Yes, you can actually create geometric spaces in which angles can be imaginary and even complex.
The trick is to note that the most general definition of angle is staged within the framework of inner product spaces: vector spaces which have an operation called the "inner product" that in effect defines how a "dot product" works for that type of vector. This definition is based on how that, for vectors in Euclidean $n$-dimensional space, we can define the angle between them via
$$\theta(\mathbf{v}, \mathbf{w}) := \cos^{-1}\left(\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}| |\mathbf{w}|}\right)$$
which, in turn, is derived from the "geometric dot product"
$$\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}| |\mathbf{w}| \cos \theta$$
where $\theta$ is the angle between the vectors, but we now go the other way and take the dot product as a primitive operation, not length and angle, and we use that to define length and angle, which are concepts that don't exist in an ordinary vector space. (Thus, we should note also that the length of $|\mathbf{v}|$ is then defined by, as one might think, $|\mathbf{v}| := \sqrt{\mathbf{v} \cdot \mathbf{v}}$.)
To see how this framework works, note that for real, say 2-dimensional, Euclidean vector space, we have that vectors look like
$$\mathbf{v} = \langle v_x, v_y \rangle$$
so that
$$\mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y$$
and from which we get the usual angle between two points in 2-dimensional space.
However, because we are talking more general vector spaces, there is no need for the scalar components of a vector to be real numbers only: we can - and do! - just as well take them as complex numbers. Of course, if we have vectors of two complex numbers, that is effectively like having four real-number dimensions, but we are defining a new kind of geometry on that. In particular, if we now have two-dimensional vectors $\mathbf{v}$ with the same form as given, only components $v_x$ and $v_y$ are complex, so that the space has two complex dimensions (and so four real dimensions), we can define an inner product like
$$\mathbf{v} \cdot \mathbf{w} := v_x \bar{w}_x + v_y \bar{w}_y$$
where you note we take the conjugate: this is to ensure that in various ways the dot product is "nice" to the structure of the complex numbers. Moreover, it ensures that the length of a vector continues to remain real (exercise: check this by considering what $\mathbf{v} \cdot \mathbf{v}$ is and then think of a certain fundamental property of the complex numbers).
So then if we do this, and we return to our definition of $\theta(\mathbf{v}, \mathbf{w})$, we can see that while the denominator of what's inside $\cos^{-1}$ is just lengths and so will be real, the numerator, which is a bare inner product, has no need to be real at all, and so we can in general get an inverse cosine of a complex number and so find vectors with an angle which is complex.
