## 1 Introduction  
In 1977, [PERSON] and [PERSON] [1], while studying first-order non-linear delay differential equations to describe physical control systems, first discovered that chaos exists in delay systems. Following this, researchers began to pay close attention to time delay chaotic systems, finding that those with a time delay can better describe real physical processes than systems without a time delay. On the basis of the stability of a class of linear-system theory, the synchronization of Chen systems with time-varying delay has been discussed. Research on time delay chaotic systems with real variables, as well as on time delay chaotic systems with complex variables, has made great progress. Since [PERSON] et al. [2] introduced the complex Lorenz system, research on the characteristics of complex chaotic systems and their synchronization has attracted much attention [3; 4; 5; 6; 7; 8]. However, time delay is an inevitable factor in actual complex chaotic systems. Therefore, many scholars have studied complex chaotic systems with time delay [9; 10; 11]. Since [PERSON] realized chaotic synchronization by circuits [12], the application of chaotic systems in secure communication has become a hot issue in the field of information security. Similarly to random signals, chaotic signals have very complex trajectories, are difficult to predict, and have inherent concealment, which make them suitable as carriers for secure communication [13; 14; 15; 16; 17]. Generally speaking, the information is encrypted as streams with chaotic characteristics at the sending end. At the receiving end, the correct information is decrypted from the received signal. Chaotic secure communication requires the synchronization of chaotic systems at the sending and receiving ends. A chaotic system is completely determined by a non-linear system of equations, including its parameters and initial conditions. The most interesting fact is that a determinate system can produce a large number of complex, uncorrelated, and quasi-random chaotic sequences. Time delay is inevitable in nature, where time delay systems have more complicated dynamic characteristics than ordinary chaotic systems. Therefore, time delay complex chaotic systems have become a prominent topic in current chaotic secure communication research. Countless scholars have conducted research on secure communication, in order to achieve high security levels. In order to achieve fast communication, [PERSON] and [PERSON] used the proportional characteristics of projection synchronization (PS) and extended binary to M-ary (\(M>2\)) systems. [PERSON]. [18,19] studied the projection synchronization of complex hyperchaotic systems and proposed corresponding communication schemes. As the rapid communication of projection synchronization and the unpredictability of the scale factor can increase communication security, PS has been widely used. On the basis of the definition of real scale-factor projection synchronization and modified projection synchronization (MPS) [20,21], the definition of modified projection synchronization with a complex scale factor--in fact, complex modified projection synchronization (CMPS)--has been proposed by [PERSON] et al. [22] and [PERSON] [23], almost at the same time. If the real part of a complex scale factor is 1 and its imaginary part is 0, it is complete synchronization (CS); if the real part of the complex scale factor is \(-1\) and the imaginary part is 0, it is the anti synchronization (AS); if the real part of the complex scale factor is a real number and the imaginary part is 0, it is the projection synchronization (PS) of the real scale factor; if only the imaginary part is 0, it is the modified projection synchronization (MPS) of the real scale factor; if the complex scale factor is \(j\), it is complex complete synchronization (CCS). Therefore, CS, AS, PS, MPS, and CCS are all special cases of CMPS.