Suppose I use the Cayley-Dickson construction on complex numbers to obtain the quaternions. I then have the "real" and the "imaginary" parts of the quaternions.
The argument of a complex number $x$ is then $\mathrm{atan2}(\mathrm{Im}(x), \mathrm{Re}(x))$ where $\mathrm{atan2}$ is like the regular arctangent function but takes into account the quadrant where the angle lies.
Shouldn't the argument of a quarternion $x$ then be a complex number that is something like $\mathrm{catan}\left(\frac{\mathrm{Im}(x)}{\mathrm{Re}(x)}\right)$ where $\mathrm{catan}$ extends the arctangent to the complex plane? The trouble I'm having here is that I need to take into account the proper "quadrants" or actually octants and I don't know how that'd work.
Given that the logarithm of a complex number $r \times e^{i \theta}$ is $\ln r + i \theta$ (plus all the other angles) the logarithm of a quaternion $x$ should be $\ln \left\lvert x \right\rvert + ix_\theta$. However, $\ln \left\lvert x \right\rvert$ for a quaternion is purely real (not just "real") and it has no $i$, $j$, or $k$ components. So the straightforward generalization of the argument misses out one one component of the quaternion.
Basically, to make the math nice the argument of a quaternion shouldn't be a complex number but instead a quaternion lacking a real component. I suspect that these three numbers would be the Euler angles. There should really be a name for this one particular case.
It takes $4$ numbers to specify a quaternion whether you use Cartesian or polar coordinates, so the phase will have 3 parts forming a quaternion of zero real part, as you guessed. Suppose a quaternion $z$ is written as $z=re^q$ where $\Re q = 0$. Identifying $bi+cj+dk$ with the vector $b\mathbf{i}+c\mathbf{j}+d\mathbf{k}\in\mathbb{R}^3$, we have $q^2=-\mathbf{q}^2$ since $ij=-ji$ etc. Taylor series then give$$e^q =\cosh q+q\text{sinch} q = \cos\left|\mathbf{q}\right|+q\text{sinc}\left|\mathbf{q}\right|,$$which is a quaternion of square modulus since $\cos^2\left|\mathbf{q}\right|+\sin^2\left|\mathbf{q}\right|=1$ as required. (We can also use this technique to give polar forms of other numbers in the Cayley-Dickson construction, such as octonions.)
A quaternion can be written in the form $a+b\hat{q}$ with $a,\,b\in\mathbb{R},\,\hat{q}^2=-1$. This is $\sqrt{a^2+b^2}\exp \text{atan2}\left( b,\,a \right)\hat{q}$.