Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly approximable by rationals. But I wonder if there is another sense in which one number is more irrational than another.

Consider the following well known irrationals: 
$\sqrt{2}$,
$\varphi$,
$\log_2{3}$,
$e$,
$\pi$,
$\zeta(3)$.

The proofs of irrationality of these numbers increase in difficulty from grade-school arguments, to calculus, to advanced methods. Other probable irrationals such as $\gamma$ most likely have very difficult proofs.

Can this notion be made precise? Is there a well defined way in which, for example, $\pi$ is more irrational than $e?$

Yes, there is such a thing as the irrationality measure of a real number (I'm not sure if it can be / has already been extended to complex numbers). It is based on the idea that all algebraic numbers (including the golden ratio) are hard to approximate well by rationals, relative to the size of the denominator of the rational used, while it is sometimes possible for a transcendental number to be approximated better. In particular, if a number $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ has the property that there are infinitely many rational approximations $\frac pq\in\mathbb{Q}$ with $|\,\alpha-\frac pq| < q^{-t}$, then $t$ is a lower bound for the irrationality measure of $\alpha$; the larger $t$ is, i.e. the better your approximations are relative to the denominator, the "more irrational" you are, at least from a Diophantine approximation point of view.

From Wikipedia: The irrationality measure of a rational number is 1; the very deep theorem of Thue, Siegel, and Roth shows that any algebraic number that isn't rational has irrationality measure 2; and transcendental numbers will have an irrationality measure $\geq2$. However, as Douglas Zare has pointed out in the comments, the set of transcendental numbers of irrationality measure $>2$ has measure 0, so that in most cases it's unfortunately not useful as a comparison.

It appears that the irrationality measure of $\pi$ is not currently known, but that there are upper bounds; the most recent one I could find is this, which would appear to show that $\mu(\pi)\leq7.6063$. The Wikipedia article claims that $\mu(e)=2$, so whether or not $\pi$ is "more irrational" than $e$ looks like an open question.    
