Hamiltonian non intersecting path in plane
Hamiltonian path problem remains NP-complete in planar graphs [1], so your problem is also NP-complete since in a planar graph, two edges cannot intersect with each other.

[1] Garey, M. R., Johnson, D. S., &amp; Tarjan, R. E. (1976). The planar Hamiltonian circuit problem is NP-complete. SIAM Journal on Computing, 5(4), 704-714.