Is there any man worth dying for? What level of love will cause a woman to commit suicide?

Jesus thought so, and acted accordingly. If someone is in danger and needs to be rescued/saved. I can think of people I would risk my life for.

About four times as many men die by suicide as women, so I find the apparent assumptions built to the question rather troublesome. Regardless, committing suicide is universally considered caused by a mental illness, so there is no such level of love that alone would cause one to commit suicide.
It can of course be a contributing factor, along with unemployment, debt, movies, examples of suicide etc. and while some people might commit suicide because they feel they've never been loved, others will throw away the perfect love and only feel emboldened by being independent and strong. Of course, if you look for the poetic/romantic answer, Romeo & Juliet I guess ;)

If that man is that woman's son. A mother's love for her children is very strong. I'm just coming to realize this.

There is no man worth dying for. If he cheats or leaves, goodbye. If he's abusive or neglectful, bye and good riddance. I don't make my entire existence about a man. My children always come first. I can love a man, want a man, but if he leaves, I will still be strong and carry on.

Is that love I'm not sure? That is the story of what Jesus supposedly did. There is deeper meaning than what the minds of the past interpreters have led us to believe.

When he chases her.

I would for my husband.