I would appreciate feedback, however, I would probably think of this in two steps to make it easier. First of all, there aren't any "imaginary" signals...if we think of voltage, in the end we are only concerned about the positive or negative, real value of the voltage, which in alternating current most typically has a sinodial form, let's assume it's some function U(t) = cos (wt).

1. In many instances it is easier to work in the exponential form as opposed to the trigonometric form, so we like transforming our signal into the exponential realm, which happens to require the complex representation. As per the Euler formula we know that exp(wt) = cos (wt) + j sin(wt). That being said, we still have to add this imaginary sin component in order to transfer this into the exponential form. In order to again get our voltage, we'd only be concerned about the real component of this, hence U(t) = Re{exp(wt)} = Re {cos (wt) + j sin (wt)} = cos(wt).

2. If I now were to take the signal j * exp(wt), so an imaginary signal we'd get: j * exp(wt) = j cos(wt) + j * j * sin (wt) = j cos(wt) - sin (wt). However, I am again only concerned about the real component of this which is: Re{j cos(wt) - sin(w)} = - sin(wt) = cos (wt + 90°).

As such the imaginary signal represents it's real counterpart shifted by 90 degrees.

I would appreciate feedback, however, I would probably think of this in two steps to make it easier. First of all, there aren't any "imaginary" signals...if we think of voltage, in the end we are only concerned about the positive or negative, real value of the voltage, which in alternating current most typically has a sinodial form, let's assume it's some function U(t) = cos (wt).

1. In many instances it is easier to work in the exponential form as opposed to the trigonometric form, so we like transforming our signal into the exponential realm, which happens to require the complex representation. As per the Euler formula we know that exp(jwt) = cos (wt) + j sin(wt). That being said, we still have to add this imaginary sin component in order to transfer this into the exponential form. In order to again get our voltage, we'd only be concerned about the real component of this, hence U(t) = Re{exp(jwt)} = Re {cos (wt) + j sin (wt)} = cos(wt).

2. If I now were to take the signal j * exp(jwt), so an imaginary signal we'd get: j * exp(jwt) = j cos(wt) + j * j * sin (wt) = j cos(wt) - sin (wt). However, I am again only concerned about the real component of this which is: Re{j cos(wt) - sin(w)} = - sin(wt) = cos (wt + 90°).

As such the imaginary signal represents it's real counterpart shifted by 90 degrees.