Block diagram for a complex impulse response
A digital system with complex input and complex output has a block diagram in which the various arrows carry complex signals, each register holds a complex number, each multiplier computes the product of two complex numbers, each adder computes the sum of two complex signals, etc. Consider, for illustration, a simple FIR filter with complex input $x$ and complex output $y$ where
$$y[n] = x[n] + a\cdot x[n-1]\tag{1}$$
with $a$ being a complex constant.  The corresponding block diagram is as shown below:


But, in real life, complex-valued signals actually consist of two separate real-valued signals traveling on two separate wires and it is only in our minds that we think these two real-valued signals as constituting a complex-valued signal (In software or MATLAB, each complex sample is stored as a pair of real numbers (two words)).  So, if we express complex input signal $x$ as $x=x_r+jx_i$ and complex output signal $y$ as $y= y_r + jy_i$ where $x_r, x_i, y_r, y_i$ are real-valued signals, what we get are  two circuits each with separate (real-valued) input and separate real-valued output, and internal cross-connections between the two circuits. We have that
\begin{align}
y[n] &amp;= x[n] + a\cdot x[n-1]\\
&amp;\Downarrow\\
y_r[n] + jy_i[n] &amp;= x_r[n] + jx_i[n] 
+ \big(a_r +ja_i\big)\big(x_r[n-1]+jx_i[n-1\big])\\
&amp;\Downarrow\\
y_r[n] &amp;= x_r[n] + a_rx_r[n-1]-a_ix_i[n-1]\tag{2}\\
y_i[n] &amp;= x_i[n] + a_rx_i[n-1] + a_ix_r[n-1]\tag{3}
\end{align}
We see that the circuit with input $x_r$ has output $y_r$ that depends on its inputs $x_r[n]$ and $x_r[n-1]$ (as it should) in exactly the same way as $(1)$ but also includes the cross-connection term $-\alpha_ix_i[n-1]$ which depends on $x_i$, the input to the other circuit. Similarly, the circuit with input $x_i$ has output $y_i$ which depends not just on $x_i[n]$ and $x_i[n-1]$ (as it should) in exactly the same way as $(1)$ but also includes the cross-connection term $\alpha_i x_r[n-1]$ which depends on $x_r$, the input to the other circuit. Thus, the block diagram above can be duplicated with some minor changes to labels (e.g. $x[n]$ is replaced by $x_r[n]$ in one circuit and by $x_i[n]$ in the other circuit, and similarly for $x[n-1]$, $a$ and $y[n]$) and the cross-connections added.  I will leave the actual task of sketching the full diagram to those interested in crossing every i and dotting every t.

And that's it.  No messing around with $\sin()$ and $\cos()$ or orthogonality, no phase-shifting by $\pi/2$, or time-shifting